Simplify the expression using a formula. Tasks for independent solution. Parenthesizing the multiplier

Simplifying algebraic expressions is one of the keys to learning algebra and an extremely useful skill for all mathematicians. Simplification allows you to reduce a complex or long expression to a simple expression that is easy to work with. Basic simplification skills are good even for those who are not enthusiastic about mathematics. By following a few simple rules, many of the most common types of algebraic expressions can be simplified without any special mathematical knowledge.

Steps

Important Definitions

1. Similar members. These are members with a variable of the same order, members with the same variables, or free members (members that do not contain a variable). In other words, like terms include one variable to the same extent, include several identical variables, or do not include a variable at all. The order of the terms in the expression does not matter.

   • For example, 3x 2 and 4x 2 are like terms because they contain the variable "x" of the second order (in the second power). However, x and x 2 are not similar members, since they contain the variable "x" of different orders (first and second). Similarly, -3yx and 5xz are not similar members because they contain different variables.

2. Factorization. This is finding such numbers, the product of which leads to the original number. Any original number can have several factors. For example, the number 12 can be decomposed into the following series of factors: 1 × 12, 2 × 6 and 3 × 4, so we can say that the numbers 1, 2, 3, 4, 6 and 12 are factors of the number 12. The factors are the same as divisors , that is, the numbers by which the original number is divisible.

   • For example, if you want to factor the number 20, write it like this: 4×5.
   • Note that when factoring, the variable is taken into account. For example, 20x = 4(5x).
   • Prime numbers cannot be factored because they are only divisible by themselves and 1.

3. Remember and follow the order of operations to avoid mistakes.

   • Brackets
   • Degree
   • Multiplication
   • Division
   • Addition
   • Subtraction

   Casting Like Members

   1. Write down the expression. The simplest algebraic expressions (which do not contain fractions, roots, and so on) can be solved (simplified) in just a few steps.

      • For example, simplify the expression 1 + 2x - 3 + 4x.

   2. Define similar members (members with a variable of the same order, members with the same variables, or free members).

      • Find similar terms in this expression. The terms 2x and 4x contain a variable of the same order (first). Also, 1 and -3 are free members (do not contain a variable). Thus, in this expression, the terms 2x and 4x are similar, and the members 1 and -3 are also similar.

   3. Give similar members. This means adding or subtracting them and simplifying the expression.

      • 2x+4x= 6x
      • 1 - 3 = -2

   4. Rewrite the expression taking into account the given terms. You will get a simple expression with fewer terms. The new expression is equal to the original.

      • In our example: 1 + 2x - 3 + 4x = 6x - 2, that is, the original expression is simplified and easier to work with.

   5. Observe the order in which operations are performed when casting like terms. In our example, it was easy to bring similar terms. However, in the case of complex expressions in which members are enclosed in brackets and fractions and roots are present, it is not so easy to bring such terms. In these cases, follow the order of operations.

      • For example, consider the expression 5(3x - 1) + x((2x)/(2)) + 8 - 3x. Here it would be a mistake to immediately define 3x and 2x as like terms and quote them, because first you need to expand the parentheses. Therefore, perform the operations in their order.
        • 5(3x-1) + x((2x)/(2)) + 8 - 3x
        • 15x - 5 + x(x) + 8 - 3x
        • 15x - 5 + x 2 + 8 - 3x. Now, when the expression contains only addition and subtraction operations, you can cast like terms.
        • x 2 + (15x - 3x) + (8 - 5)
        • x 2 + 12x + 3

   Parenthesizing the multiplier

   1. Find the greatest common divisor (gcd) of all coefficients of the expression. GCD is the largest number by which all coefficients of the expression are divisible.

      • For example, consider the equation 9x 2 + 27x - 3. In this case, gcd=3, since any coefficient of this expression is divisible by 3.

   2. Divide each term of the expression by gcd. The resulting terms will contain smaller coefficients than in the original expression.

      • In our example, divide each expression term by 3.
        • 9x2/3=3x2
        • 27x/3=9x
        • -3/3 = -1
        • It turned out the expression 3x2 + 9x-1. It is not equal to the original expression.

   3. Write the original expression as equal to the product of gcd times the resulting expression. That is, enclose the resulting expression in brackets, and put the GCD out of brackets.

      • In our example: 9x 2 + 27x - 3 = 3(3x 2 + 9x - 1)

   4. Simplifying fractional expressions by taking the multiplier out of brackets. Why just take the multiplier out of brackets, as was done earlier? Then, to learn how to simplify complex expressions, such as fractional expressions. In this case, putting the factor out of the brackets can help get rid of the fraction (from the denominator).

      • For example, consider the fractional expression (9x 2 + 27x - 3)/3. Use parentheses to simplify this expression.
        • Factor out the factor 3 (as you did before): (3(3x 2 + 9x - 1))/3
        • Note that both the numerator and denominator now have the number 3. This can be reduced, and you get the expression: (3x 2 + 9x - 1) / 1
        • Since any fraction that has the number 1 in the denominator is just equal to the numerator, the original fractional expression is simplified to: 3x2 + 9x-1.

   Additional Simplification Techniques

4. Consider a simple example: √(90). The number 90 can be decomposed into the following factors: 9 and 10, and from 9, take the square root (3) and take 3 out from under the root.
   • √(90)
   • √(9×10)
   • √(9)×√(10)
   • 3×√(10)
   • 3√(10)

5. Simplifying expressions with powers. In some expressions, there are operations of multiplication or division of terms with a degree. In the case of multiplication of terms with one base, their degrees are added; in the case of dividing terms with the same base, their degrees are subtracted.

   • For example, consider the expression 6x 3 × 8x 4 + (x 17 / x 15). In the case of multiplication, add the exponents, and in the case of division, subtract them.
     • 6x 3 × 8x 4 + (x 17 / x 15)
     • (6 × 8)x 3 + 4 + (x 17 - 15)
     • 48x7+x2
   • The following is an explanation of the rule for multiplying and dividing terms with a degree.
     • Multiplying terms with powers is equivalent to multiplying terms by themselves. For example, since x 3 = x × x × x and x 5 = x × x × x × x × x, then x 3 × x 5 = (x × x × x) × (x × x × x × x × x), or x 8 .
     • Similarly, dividing terms with powers is equivalent to dividing terms by themselves. x 5 /x 3 \u003d (x × x × x × x × x) / (x × x × x). Since similar terms that are in both the numerator and the denominator can be reduced, the product of two "x", or x 2, remains in the numerator.

• Always be aware of the signs (plus or minus) in front of the terms of an expression, as many people have difficulty choosing the right sign.
• Ask for help if needed!
• Simplifying algebraic expressions is not easy, but if you get your hands on it, you can use this skill for a lifetime.

Among the various expressions that are considered in algebra, sums of monomials occupy an important place. Here are examples of such expressions:
\(5a^4 - 2a^3 + 0.3a^2 - 4.6a + 8 \)
\(xy^3 - 5x^2y + 9x^3 - 7y^2 + 6x + 5y - 2 \)

The sum of monomials is called a polynomial. The terms in a polynomial are called members of the polynomial. Mononomials are also referred to as polynomials, considering a monomial as a polynomial consisting of one member.

For example, polynomial
\(8b^5 - 2b \cdot 7b^4 + 3b^2 - 8b + 0.25b \cdot (-12)b + 16 \)
can be simplified.

We represent all the terms as monomials of the standard form:
\(8b^5 - 2b \cdot 7b^4 + 3b^2 - 8b + 0.25b \cdot (-12)b + 16 = \)
\(= 8b^5 - 14b^5 + 3b^2 -8b -3b^2 + 16 \)

We give similar terms in the resulting polynomial:
\(8b^5 -14b^5 +3b^2 -8b -3b^2 + 16 = -6b^5 -8b + 16 \)
The result is a polynomial, all members of which are monomials of the standard form, and among them there are no similar ones. Such polynomials are called polynomials of standard form.

Behind polynomial degree standard form take the largest of the powers of its members. So, the binomial \(12a^2b - 7b \) has the third degree, and the trinomial \(2b^2 -7b + 6 \) has the second.

Usually, the terms of standard form polynomials containing one variable are arranged in descending order of its exponents. For example:
\(5x - 18x^3 + 1 + x^5 = x^5 - 18x^3 + 5x + 1 \)

The sum of several polynomials can be converted (simplified) into a standard form polynomial.

Sometimes the members of a polynomial need to be divided into groups, enclosing each group in parentheses. Since parentheses are the opposite of parentheses, it is easy to formulate parentheses opening rules:

If the + sign is placed before the brackets, then the terms enclosed in brackets are written with the same signs.

If a "-" sign is placed in front of the brackets, then the terms enclosed in brackets are written with opposite signs.

Transformation (simplification) of the product of a monomial and a polynomial

Using the distributive property of multiplication, one can transform (simplify) the product of a monomial and a polynomial into a polynomial. For example:
\(9a^2b(7a^2 - 5ab - 4b^2) = \)
\(= 9a^2b \cdot 7a^2 + 9a^2b \cdot (-5ab) + 9a^2b \cdot (-4b^2) = \)
\(= 63a^4b - 45a^3b^2 - 36a^2b^3 \)

The product of a monomial and a polynomial is identically equal to the sum of the products of this monomial and each of the terms of the polynomial.

This result is usually formulated as a rule.

To multiply a monomial by a polynomial, one must multiply this monomial by each of the terms of the polynomial.

We have repeatedly used this rule for multiplying by a sum.

The product of polynomials. Transformation (simplification) of the product of two polynomials

In general, the product of two polynomials is identically equal to the sum of the product of each term of one polynomial and each term of the other.

Usually use the following rule.

To multiply a polynomial by a polynomial, you need to multiply each term of one polynomial by each term of the other and add the resulting products.

Abbreviated multiplication formulas. Sum, Difference, and Difference Squares

Some expressions in algebraic transformations have to be dealt with more often than others. Perhaps the most common expressions are \((a + b)^2, \; (a - b)^2 \) and \(a^2 - b^2 \), that is, the square of the sum, the square of the difference, and square difference. You have noticed that the names of these expressions seem to be incomplete, so, for example, \((a + b)^2 \) is, of course, not just the square of the sum, but the square of the sum of a and b. However, the square of the sum of a and b is not so common, as a rule, instead of the letters a and b, it contains various, sometimes quite complex expressions.

Expressions \((a + b)^2, \; (a - b)^2 \) are easy to convert (simplify) into polynomials of the standard form, in fact, you have already met with such a task when multiplying polynomials:
\((a + b)^2 = (a + b)(a + b) = a^2 + ab + ba + b^2 = \)
\(= a^2 + 2ab + b^2 \)

The resulting identities are useful to remember and apply without intermediate calculations. Short verbal formulations help this.

\((a + b)^2 = a^2 + b^2 + 2ab \) - the square of the sum is equal to the sum of the squares and the double product.

\((a - b)^2 = a^2 + b^2 - 2ab \) - the square of the difference is the sum of the squares without doubling the product.

\(a^2 - b^2 = (a - b)(a + b) \) - the difference of squares is equal to the product of the difference and the sum.

These three identities allow in transformations to replace their left parts with right ones and vice versa - right parts with left ones. The most difficult thing in this case is to see the corresponding expressions and understand what the variables a and b are replaced in them. Let's look at a few examples of using abbreviated multiplication formulas.

It is known that in mathematics one cannot do without simplifying expressions. This is necessary for the correct and quick solution of a wide variety of problems, as well as various kinds of equations. The discussed simplification implies a reduction in the number of actions necessary to achieve the goal. As a result, calculations are noticeably facilitated, and time is significantly saved. But how to simplify the expression? For this, established mathematical relationships are used, often referred to as formulas, or laws that allow you to make expressions much shorter, thereby simplifying calculations.

It's no secret that today it's not difficult to simplify the expression online. Here are links to some of the more popular ones:

However, this is not possible with every expression. Therefore, we will consider more traditional methods in more detail.

Taking out a common divisor

In the case when in one expression there are monomials that have the same factors, you can find the sum of the coefficients with them, and then multiply by the common factor for them. This operation is also called "subtracting a common divisor". Consistently using this method, sometimes you can significantly simplify the expression. Algebra, after all, in general, as a whole, is built on the grouping and regrouping of factors and divisors.

The simplest formulas for abbreviated multiplication

One of the consequences of the previously described method are the reduced multiplication formulas. How to simplify expressions with their help is much clearer to those who have not even learned these formulas by heart, but know how they are derived, that is, where they come from, and, accordingly, their mathematical nature. In principle, the previous statement remains valid in all modern mathematics, from the first grade to the higher courses of the Mechanics and Mathematics departments. The difference of squares, the square of the difference and the sum, the sum and difference of cubes - all these formulas are widely used in elementary, as well as higher mathematics, in cases where it is necessary to simplify the expression to solve the problems. Examples of such transformations can be easily found in any school textbook on algebra, or, even simpler, on the vastness of the worldwide web.

Degree roots

Elementary mathematics, if you look at it as a whole, is armed with not so many ways in which you can simplify the expression. Degrees and actions with them, as a rule, are relatively easy for most students. Only now, many modern schoolchildren and students have considerable difficulties when it is necessary to simplify the expression with roots. And it's completely unfounded. Because the mathematical nature of the roots is no different from the nature of the same degrees, with which, as a rule, there are much fewer difficulties. It is known that the square root of a number, variable or expression is nothing but the same number, variable or expression to the power of "one second", the cube root is the same to the power of "one third", and so on by correspondence.

Simplifying expressions with fractions

Consider also a common example of how to simplify an expression with fractions. In cases where the expressions are natural fractions, a common factor should be extracted from the denominator and numerator, and then the fraction should be reduced by it. When the monomials have the same multipliers raised to powers, it is necessary to monitor the equality of the powers when summing them.

Simplification of the simplest trigonometric expressions

Some apart is the conversation about how to simplify the trigonometric expression. The broadest section of trigonometry is, perhaps, the first stage at which students of mathematics will encounter somewhat abstract concepts, problems and methods for solving them. Here there are corresponding formulas, the first of which is the basic trigonometric identity. Having a sufficient mathematical mindset, one can trace the systematic derivation from this identity of all the main trigonometric identities and formulas, including formulas for the difference and sum of arguments, double, triple arguments, reduction formulas and many others. Of course, one should not forget here the very first methods, such as taking out a common factor, which are fully used along with new methods and formulas.

To summarize, here are some general tips for the reader:

• Polynomials should be factored, that is, they should be represented in the form of a product of a certain number of factors - monomials and polynomials. If there is such a possibility, it is necessary to take the common factor out of brackets.
• It’s better to memorize all the abbreviated multiplication formulas without exception. There are not so many of them, but they are the basis for simplifying mathematical expressions. You should also not forget about the method of highlighting perfect squares in trinomials, which is the inverse action to one of the abbreviated multiplication formulas.
• All existing fractions in the expression should be reduced as often as possible. In doing so, do not forget that only multipliers are reduced. In the case when the denominator and numerator of algebraic fractions are multiplied by the same number, which differs from zero, the values ​​of the fractions do not change.
• In general, all expressions can be transformed by actions, or by a chain. The first method is more preferable, because. the results of intermediate actions are more easily verified.
• Quite often, in mathematical expressions, you have to extract the roots. It should be remembered that the roots of even degrees can be extracted only from a non-negative number or expression, and the roots of odd degrees can be extracted completely from any expressions or numbers.

We hope that our article will help you, in the future, to understand mathematical formulas and teach you how to apply them in practice.

Let's consider the topic of transforming expressions with powers, but first we will dwell on a number of transformations that can be carried out with any expressions, including power ones. We will learn how to open brackets, give like terms, work with the base and exponent, use the properties of degrees.

Yandex.RTB R-A-339285-1

What are Power Expressions?

In the school course, few people use the phrase "power expressions", but this term is constantly found in collections for preparing for the exam. In most cases, the phrase denotes expressions that contain degrees in their entries. This is what we will reflect in our definition.

Definition 1

Power expression is an expression that contains degrees.

We give several examples of power expressions, starting with a degree with a natural exponent and ending with a degree with a real exponent.

The simplest power expressions can be considered powers of a number with a natural exponent: 3 2 , 7 5 + 1 , (2 + 1) 5 , (− 0 , 1) 4 , 2 2 3 3 , 3 a 2 − a + a 2 , x 3 − 1 , (a 2) 3 . As well as powers with zero exponent: 5 0 , (a + 1) 0 , 3 + 5 2 − 3 , 2 0 . And powers with negative integer powers: (0 , 5) 2 + (0 , 5) - 2 2 .

It is a little more difficult to work with a degree that has rational and irrational exponents: 264 1 4 - 3 3 3 1 2 , 2 3 , 5 2 - 2 2 - 1 , 5 , 1 a 1 4 a 1 2 - 2 a - 1 6 · b 1 2 , x π · x 1 - π , 2 3 3 + 5 .

The indicator can be a variable 3 x - 54 - 7 3 x - 58 or a logarithm x 2 l g x − 5 x l g x.

We have dealt with the question of what power expressions are. Now let's take a look at their transformation.

The main types of transformations of power expressions

First of all, we will consider the basic identity transformations of expressions that can be performed with power expressions.

Example 1

Calculate Power Expression Value 2 3 (4 2 − 12).

Solution

We will carry out all transformations in compliance with the order of actions. In this case, we will start by performing the actions in brackets: we will replace the degree with a digital value and calculate the difference between the two numbers. We have 2 3 (4 2 − 12) = 2 3 (16 − 12) = 2 3 4.

It remains for us to replace the degree 2 3 its meaning 8 and calculate the product 8 4 = 32. Here is our answer.

Answer: 2 3 (4 2 − 12) = 32 .

Example 2

Simplify expression with powers 3 a 4 b − 7 − 1 + 2 a 4 b − 7.

Solution

The expression given to us in the condition of the problem contains similar terms, which we can bring: 3 a 4 b − 7 − 1 + 2 a 4 b − 7 = 5 a 4 b − 7 − 1.

Answer: 3 a 4 b − 7 − 1 + 2 a 4 b − 7 = 5 a 4 b − 7 − 1 .

Example 3

Express an expression with powers of 9 - b 3 · π - 1 2 as a product.

Solution

Let's represent the number 9 as a power 3 2 and apply the abbreviated multiplication formula:

9 - b 3 π - 1 2 = 3 2 - b 3 π - 1 2 = = 3 - b 3 π - 1 3 + b 3 π - 1

Answer: 9 - b 3 π - 1 2 = 3 - b 3 π - 1 3 + b 3 π - 1 .

And now let's move on to the analysis of identical transformations that can be applied specifically to power expressions.

Working with base and exponent

The degree in the base or exponent can have numbers, variables, and some expressions. For example, (2 + 0 , 3 7) 5 − 3 , 7 And . It is difficult to work with such records. It is much easier to replace the expression in the base of the degree or the expression in the exponent with an identically equal expression.

The transformations of the degree and the indicator are carried out according to the rules known to us separately from each other. The most important thing is that as a result of the transformations, an expression is obtained that is identical to the original one.

The purpose of transformations is to simplify the original expression or to obtain a solution to the problem. For example, in the example we gave above, (2 + 0 , 3 7) 5 − 3 , 7 you can perform operations to go to the degree 4 , 1 1 , 3 . Opening the brackets, we can bring like terms in the base of the degree (a (a + 1) − a 2) 2 (x + 1) and get a power expression of a simpler form a 2 (x + 1).

Using Power Properties

The properties of degrees, written as equalities, are one of the main tools for transforming expressions with degrees. We present here the main ones, considering that a And b are any positive numbers, and r And s- arbitrary real numbers:

Definition 2

• a r a s = a r + s ;
• a r: a s = a r − s ;
• (a b) r = a r b r ;
• (a: b) r = a r: b r ;
• (a r) s = a r s .

In cases where we are dealing with natural, integer, positive exponents, the restrictions on the numbers a and b can be much less stringent. So, for example, if we consider the equality a m a n = a m + n, Where m And n are natural numbers, then it will be true for any values ​​of a, both positive and negative, as well as for a = 0.

You can apply the properties of degrees without restrictions in cases where the bases of the degrees are positive or contain variables whose range of acceptable values ​​is such that the bases take only positive values ​​on it. In fact, within the framework of the school curriculum in mathematics, the task of the student is to choose the appropriate property and apply it correctly.

When preparing for admission to universities, there may be tasks in which inaccurate application of properties will lead to a narrowing of the ODZ and other difficulties with the solution. In this section, we will consider only two such cases. More information on the subject can be found in the topic "Transforming expressions using exponent properties".

Example 4

Represent the expression a 2 , 5 (a 2) - 3: a - 5 , 5 as a degree with a base a.

Solution

To begin with, we use the exponentiation property and transform the second factor using it (a 2) − 3. Then we use the properties of multiplication and division of powers with the same base:

a 2 , 5 a − 6: a − 5 , 5 = a 2 , 5 − 6: a − 5 , 5 = a − 3 , 5: a − 5 , 5 = a − 3 , 5 − (− 5 , 5) = a 2 .

Answer: a 2 , 5 (a 2) − 3: a − 5 , 5 = a 2 .

The transformation of power expressions according to the property of degrees can be done both from left to right and in the opposite direction.

Example 5

Find the value of the power expression 3 1 3 · 7 1 3 · 21 2 3 .

Solution

If we apply the equality (a b) r = a r b r, from right to left, then we get a product of the form 3 7 1 3 21 2 3 and then 21 1 3 21 2 3 . Let's add the exponents when multiplying powers with the same bases: 21 1 3 21 2 3 \u003d 21 1 3 + 2 3 \u003d 21 1 \u003d 21.

There is another way to make transformations:

3 1 3 7 1 3 21 2 3 = 3 1 3 7 1 3 (3 7) 2 3 = 3 1 3 7 1 3 3 2 3 7 2 3 = = 3 1 3 3 2 3 7 1 3 7 2 3 = 3 1 3 + 2 3 7 1 3 + 2 3 = 3 1 7 1 = 21

Answer: 3 1 3 7 1 3 21 2 3 = 3 1 7 1 = 21

Example 6

Given a power expression a 1 , 5 − a 0 , 5 − 6, enter a new variable t = a 0 , 5.

Solution

Imagine the degree a 1 , 5 How a 0 , 5 3. Using the degree property in a degree (a r) s = a r s from right to left and get (a 0 , 5) 3: a 1 , 5 - a 0 , 5 - 6 = (a 0 , 5) 3 - a 0 , 5 - 6 . In the resulting expression, you can easily introduce a new variable t = a 0 , 5: get t 3 − t − 6.

Answer: t 3 − t − 6 .

Converting fractions containing powers

We usually deal with two variants of power expressions with fractions: the expression is a fraction with a degree or contains such a fraction. All basic fraction transformations are applicable to such expressions without restrictions. They can be reduced, brought to a new denominator, work separately with the numerator and denominator. Let's illustrate this with examples.

Example 7

Simplify the power expression 3 5 2 3 5 1 3 - 5 - 2 3 1 + 2 x 2 - 3 - 3 x 2 .

Solution

We are dealing with a fraction, so we will carry out transformations in both the numerator and the denominator:

3 5 2 3 5 1 3 - 5 - 2 3 1 + 2 x 2 - 3 - 3 x 2 = 3 5 2 3 5 1 3 - 3 5 2 3 5 - 2 3 - 2 - x 2 = = 3 5 2 3 + 1 3 - 3 5 2 3 + - 2 3 - 2 - x 2 = 3 5 1 - 3 5 0 - 2 - x 2

Put a minus in front of the fraction to change the sign of the denominator: 12 - 2 - x 2 = - 12 2 + x 2

Answer: 3 5 2 3 5 1 3 - 5 - 2 3 1 + 2 x 2 - 3 - 3 x 2 = - 12 2 + x 2

Fractions containing powers are reduced to a new denominator in the same way as rational fractions. To do this, you need to find an additional factor and multiply the numerator and denominator of the fraction by it. It is necessary to select an additional factor in such a way that it does not vanish for any values ​​of the variables from the ODZ variables for the original expression.

Example 8

Bring the fractions to a new denominator: a) a + 1 a 0, 7 to the denominator a, b) 1 x 2 3 - 2 x 1 3 y 1 6 + 4 y 1 3 to the denominator x + 8 y 1 2 .

Solution

a) We choose a factor that will allow us to reduce to a new denominator. a 0 , 7 a 0 , 3 = a 0 , 7 + 0 , 3 = a , therefore, as an additional factor, we take a 0 , 3. The range of admissible values ​​of the variable a includes the set of all positive real numbers. In this area, the degree a 0 , 3 does not go to zero.

Let's multiply the numerator and denominator of a fraction by a 0 , 3:

a + 1 a 0, 7 = a + 1 a 0, 3 a 0, 7 a 0, 3 = a + 1 a 0, 3 a

b) Pay attention to the denominator:

x 2 3 - 2 x 1 3 y 1 6 + 4 y 1 3 = = x 1 3 2 - x 1 3 2 y 1 6 + 2 y 1 6 2

Multiply this expression by x 1 3 + 2 · y 1 6 , we get the sum of cubes x 1 3 and 2 · y 1 6 , i.e. x + 8 · y 1 2 . This is our new denominator, to which we need to bring the original fraction.

So we found an additional factor x 1 3 + 2 · y 1 6 . On the range of acceptable values ​​of variables x And y the expression x 1 3 + 2 y 1 6 does not vanish, so we can multiply the numerator and denominator of the fraction by it:
1 x 2 3 - 2 x 1 3 y 1 6 + 4 y 1 3 = = x 1 3 + 2 y 1 6 x 1 3 + 2 y 1 6 x 2 3 - 2 x 1 3 y 1 6 + 4 y 1 3 = = x 1 3 + 2 y 1 6 x 1 3 3 + 2 y 1 6 3 = x 1 3 + 2 y 1 6 x + 8 y 1 2

Answer: a) a + 1 a 0, 7 = a + 1 a 0, 3 a, b) 1 x 2 3 - 2 x 1 3 y 1 6 + 4 y 1 3 = x 1 3 + 2 y 1 6 x + 8 y 1 2 .

Example 9

Reduce the fraction: a) 30 x 3 (x 0, 5 + 1) x + 2 x 1 1 3 - 5 3 45 x 0, 5 + 1 2 x + 2 x 1 1 3 - 5 3, b) a 1 4 - b 1 4 a 1 2 - b 1 2.

Solution

a) Use the greatest common denominator (GCD) by which the numerator and denominator can be reduced. For the numbers 30 and 45, this is 15 . We can also reduce x 0 , 5 + 1 and on x + 2 x 1 1 3 - 5 3 .

We get:

30 x 3 (x 0 , 5 + 1) x + 2 x 1 1 3 - 5 3 45 x 0 , 5 + 1 2 x + 2 x 1 1 3 - 5 3 = 2 x 3 3 (x 0 , 5 + 1)

b) Here the presence of identical factors is not obvious. You will have to perform some transformations in order to get the same factors in the numerator and denominator. To do this, we expand the denominator using the difference of squares formula:

a 1 4 - b 1 4 a 1 2 - b 1 2 = a 1 4 - b 1 4 a 1 4 2 - b 1 2 2 = = a 1 4 - b 1 4 a 1 4 + b 1 4 a 1 4 - b 1 4 = 1 a 1 4 + b 1 4

Answer: a) 30 x 3 (x 0, 5 + 1) x + 2 x 1 1 3 - 5 3 45 x 0, 5 + 1 2 x + 2 x 1 1 3 - 5 3 = 2 · x 3 3 · (x 0 , 5 + 1) , b) a 1 4 - b 1 4 a 1 2 - b 1 2 = 1 a 1 4 + b 1 4 .

The main operations with fractions include reduction to a new denominator and reduction of fractions. Both actions are performed in compliance with a number of rules. When adding and subtracting fractions, the fractions are first reduced to a common denominator, after which actions (addition or subtraction) are performed with numerators. The denominator remains the same. The result of our actions is a new fraction, the numerator of which is the product of the numerators, and the denominator is the product of the denominators.

Example 10

Do the steps x 1 2 + 1 x 1 2 - 1 - x 1 2 - 1 x 1 2 + 1 · 1 x 1 2 .

Solution

Let's start by subtracting the fractions that are in brackets. Let's bring them to a common denominator:

x 1 2 - 1 x 1 2 + 1

Let's subtract the numerators:

x 1 2 + 1 x 1 2 - 1 - x 1 2 - 1 x 1 2 + 1 1 x 1 2 = = x 1 2 + 1 x 1 2 + 1 x 1 2 - 1 x 1 2 + 1 - x 1 2 - 1 x 1 2 - 1 x 1 2 + 1 x 1 2 - 1 1 x 1 2 = = x 1 2 + 1 2 - x 1 2 - 1 2 x 1 2 - 1 x 1 2 + 1 1 x 1 2 = = x 1 2 2 + 2 x 1 2 + 1 - x 1 2 2 - 2 x 1 2 + 1 x 1 2 - 1 x 1 2 + 1 1 x 1 2 = = 4 x 1 2 x 1 2 - 1 x 1 2 + 1 1 x 1 2

Now we multiply fractions:

4 x 1 2 x 1 2 - 1 x 1 2 + 1 1 x 1 2 = = 4 x 1 2 x 1 2 - 1 x 1 2 + 1 x 1 2

Let's reduce by a degree x 1 2, we get 4 x 1 2 - 1 x 1 2 + 1 .

Additionally, you can simplify the power expression in the denominator using the formula for the difference of squares: squares: 4 x 1 2 - 1 x 1 2 + 1 = 4 x 1 2 2 - 1 2 = 4 x - 1.

Answer: x 1 2 + 1 x 1 2 - 1 - x 1 2 - 1 x 1 2 + 1 1 x 1 2 = 4 x - 1

Example 11

Simplify the power expression x 3 4 x 2 , 7 + 1 2 x - 5 8 x 2 , 7 + 1 3 .
Solution

We can reduce the fraction by (x 2 , 7 + 1) 2. We get a fraction x 3 4 x - 5 8 x 2, 7 + 1.

Let's continue transformations of x powers x 3 4 x - 5 8 · 1 x 2 , 7 + 1 . Now you can use the power division property with the same bases: x 3 4 x - 5 8 1 x 2, 7 + 1 = x 3 4 - - 5 8 1 x 2, 7 + 1 = x 1 1 8 1 x 2 , 7 + 1 .

We pass from the last product to the fraction x 1 3 8 x 2, 7 + 1.

Answer: x 3 4 x 2 , 7 + 1 2 x - 5 8 x 2 , 7 + 1 3 = x 1 3 8 x 2 , 7 + 1 .

In most cases, it is more convenient to transfer multipliers with negative exponents from the numerator to the denominator and vice versa by changing the sign of the exponent. This action simplifies the further decision. Let's give an example: the power expression (x + 1) - 0 , 2 3 · x - 1 can be replaced by x 3 · (x + 1) 0 , 2 .

Converting expressions with roots and powers

In tasks, there are power expressions that contain not only degrees with fractional exponents, but also roots. It is desirable to reduce such expressions only to roots or only to powers. The transition to degrees is preferable, since they are easier to work with. Such a transition is especially advantageous when the DPV of the variables for the original expression allows you to replace the roots with powers without having to access the modulus or split the DPV into several intervals.

Example 12

Express the expression x 1 9 x x 3 6 as a power.

Solution

Valid range of a variable x is determined by two inequalities x ≥ 0 and x · x 3 ≥ 0 , which define the set [ 0 , + ∞) .

On this set, we have the right to move from roots to powers:

x 1 9 x x 3 6 = x 1 9 x x 1 3 1 6

Using the properties of degrees, we simplify the resulting power expression.

x 1 9 x x 1 3 1 6 = x 1 9 x 1 6 x 1 3 1 6 = x 1 9 x 1 6 x 1 1 3 6 = = x 1 9 x 1 6 x 1 18 = x 1 9 + 1 6 + 1 18 = x 1 3

Answer: x 1 9 x x 3 6 = x 1 3 .

Converting powers with variables in the exponent

These transformations are quite simple to make if you correctly use the properties of the degree. For example, 5 2 x + 1 − 3 5 x 7 x − 14 7 2 x − 1 = 0.

We can replace the product of the degree, in terms of which the sum of some variable and a number is found. On the left side, this can be done with the first and last terms on the left side of the expression:

5 2 x 5 1 − 3 5 x 7 x − 14 7 2 x 7 − 1 = 0 , 5 5 2 x − 3 5 x 7 x − 2 7 2 x = 0 .

Now let's divide both sides of the equation by 7 2 x. This expression on the ODZ of the variable x takes only positive values:

5 5 - 3 5 x 7 x - 2 7 2 x 7 2 x = 0 7 2 x , 5 5 2 x 7 2 x - 3 5 x 7 x 7 2 x - 2 7 2 x 7 2 x = 0 , 5 5 2 x 7 2 x - 3 5 x 7 x 7 x 7 x - 2 7 2 x 7 2 x = 0

Let's reduce the fractions with powers, we get: 5 5 2 x 7 2 x - 3 5 x 7 x - 2 = 0 .

Finally, the ratio of powers with the same exponents is replaced by powers of ratios, which leads to the equation 5 5 7 2 x - 3 5 7 x - 2 = 0 , which is equivalent to 5 5 7 x 2 - 3 5 7 x - 2 = 0 .

We introduce a new variable t = 5 7 x , which reduces the solution of the original exponential equation to the solution of the quadratic equation 5 · t 2 − 3 · t − 2 = 0 .

Converting expressions with powers and logarithms

Expressions containing powers and logarithms are also found in problems. Examples of such expressions are: 1 4 1 - 5 log 2 3 or log 3 27 9 + 5 (1 - log 3 5) log 5 3 . The transformation of such expressions is carried out using the approaches discussed above and the properties of logarithms, which we have analyzed in detail in the topic “Transformation of logarithmic expressions”.

If you notice a mistake in the text, please highlight it and press Ctrl+Enter

Some algebraic examples of one kind are capable of terrifying schoolchildren. Long expressions are not only intimidating, but also very difficult to calculate. Trying to immediately understand what follows and what follows, not to get confused for long. It is for this reason that mathematicians always try to simplify the “terrible” task as much as possible and only then proceed to solve it. Oddly enough, such a trick greatly speeds up the process.

Simplification is one of the fundamental points in algebra. If in simple tasks it is still possible to do without it, then more difficult to calculate examples may be “too tough”. This is where these skills come in handy! Moreover, complex mathematical knowledge is not required: it will be enough just to remember and learn how to put into practice a few basic techniques and formulas.

Regardless of the complexity of the calculations, when solving any expression, it is important follow the order of operations with numbers:

1. brackets;
2. exponentiation;
3. multiplication;
4. division;
5. addition;
6. subtraction.

The last two points can be safely swapped and this will not affect the result in any way. But adding two neighboring numbers, when next to one of them there is a multiplication sign, is absolutely impossible! The answer, if any, is wrong. Therefore, you need to remember the sequence.

The use of such

Such elements include numbers with a variable of the same order or the same degree. There are also so-called free members that do not have next to them the letter designation of the unknown.

The bottom line is that in the absence of parentheses You can simplify the expression by adding or subtracting like.

A few illustrative examples:

• 8x 2 and 3x 2 - both numbers have the same second order variable, so they are similar and when added, they are simplified to (8+3)x 2 =11x 2, while when subtracted, it turns out (8-3)x 2 =5x 2;
• 4x 3 and 6x - and here "x" has a different degree;
• 2y 7 and 33x 7 - contain different variables, therefore, as in the previous case, they do not belong to similar ones.

Factoring a Number

This little mathematical trick, if you learn how to use it correctly, will help you to cope with a tricky problem more than once in the future. And it’s easy to understand how the “system” works: a decomposition is a product of several elements, the calculation of which gives the original value. Thus, 20 can be represented as 20x1, 2x10, 5x4, 2x5x2, or some other way.

On a note: multipliers are always the same as divisors. So you need to look for a working “pair” for expansion among the numbers by which the original is divisible without a remainder.

You can perform such an operation both with free members and with digits attached to a variable. The main thing is not to lose the latter during calculations - even after decomposition, the unknown cannot take and "go nowhere." It remains at one of the factors:

• 15x=3(5x);
• 60y 2 \u003d (15y 2) 4.

Prime numbers that can only be divided by themselves or 1 never factor - it makes no sense..

Basic Simplification Methods

The first thing that catches the eye:

• the presence of brackets;
• fractions;
• roots.

Algebraic examples in the school curriculum are often compiled with the assumption that they can be beautifully simplified.

Bracket Calculations

Pay close attention to the sign in front of the brackets! Multiplication or division is applied to each element inside, and minus - reverses the existing "+" or "-" signs.

Parentheses are calculated according to the rules or according to the formulas of abbreviated multiplication, after which similar ones are given.

Fraction reduction

Reduce fractions is also easy. They themselves “willingly run away” once in a while, it is worth making operations with bringing such members. But you can simplify the example even before this: pay attention to the numerator and denominator. They often contain explicit or hidden elements that can be mutually reduced. True, if in the first case you just need to delete the superfluous, in the second you will have to think, bringing part of the expression to the form for simplification. Methods used:

• search and bracketing of the greatest common divisor of the numerator and denominator;
• dividing each top element by the denominator.

When an expression or part of it is under the root, the primary simplification problem is almost the same as the case with fractions. It is necessary to look for ways to completely get rid of it or, if this is not possible, to minimize the sign interfering with calculations. For example, to unobtrusive √(3) or √(7).

A sure way to simplify the radical expression is to try to factor it out, some of which are outside the sign. An illustrative example: √(90)=√(9×10) =√(9)×√(10)=3√(10).

Other little tricks and nuances:

• this simplification operation can be carried out with fractions, taking it out of the sign both as a whole and separately as a numerator or denominator;
• it is impossible to decompose and take out a part of the sum or difference beyond the root;
• when working with variables, be sure to take into account its degree, it must be equal to or a multiple of the root for the possibility of rendering: √(x 2 y)=x√(y), √(x 3)=√(x 2 ×x)=x√( x);
• sometimes it is allowed to get rid of the radical variable by raising it to a fractional power: √ (y 3)=y 3/2.

Power Expression Simplification

If in the case of simple calculations by minus or plus, examples are simplified by bringing similar ones, then what about when multiplying or dividing variables with different powers? They can be easily simplified by remembering two main points:

1. If there is a multiplication sign between the variables, the exponents are added.
2. When they are divided by each other, the same denominator is subtracted from the degree of the numerator.

The only condition for such a simplification is that both terms have the same basis. Examples for clarity:

• 5x 2 × 4x 7 + (y 13 / y 11) \u003d (5 × 4)x 2+7 + y 13- 11 \u003d 20x 9 + y 2;
• 2z 3 +z×z 2 -(3×z 8 /z 5)=2z 3 +z 1+2 -(3×z 8-5)=2z 3 +z 3 -3z 3 =3z 3 -3z 3 = 0.

We note that operations with numerical values ​​in front of variables occur according to the usual mathematical rules. And if you look closely, it becomes clear that the power elements of the expression "work" in a similar way:

• raising a member to a power means multiplying it by itself a certain number of times, i.e. x 2 \u003d x × x;
• division is similar: if you expand the degree of the numerator and denominator, then some of the variables will be reduced, while the rest are “gathered”, which is equivalent to subtraction.

As in any business, when simplifying algebraic expressions, not only knowledge of the basics is necessary, but also practice. After just a few lessons, examples that once seemed complicated will be reduced without much difficulty, turning into short and easily solved ones.

Video

This video will help you understand and remember how expressions are simplified.

Didn't get an answer to your question? Suggest a topic to the authors.