Power expressions (expressions with powers) and their transformation. Posts tagged "simplify algebraic expression"

Using any language, you can express the same information in different words and phrases. Mathematical language is no exception. But the same expression can be equivalently written in different ways. And in some situations, one of the entries is simpler. We'll talk about simplifying expressions in this lesson.

People communicate on different languages. For us, an important comparison is the pair “Russian language - mathematical language”. The same information can be communicated in different languages. But, besides this, it can be pronounced in different ways in one language.

For example: “Petya is friends with Vasya”, “Vasya is friends with Petya”, “Petya and Vasya are friends”. Said differently, but the same thing. From any of these phrases we would understand what we are talking about.

Let's look at this phrase: “The boy Petya and the boy Vasya are friends.” We understand what we are talking about. However, we don't like the sound of this phrase. Can't we simplify it, say the same thing, but simpler? “Boy and boy” - you can say once: “Boys Petya and Vasya are friends.”

“Boys”... Isn’t it clear from their names that they are not girls? We remove the “boys”: “Petya and Vasya are friends.” And the word “friends” can be replaced with “friends”: “Petya and Vasya are friends.” As a result, the first, long, ugly phrase was replaced with an equivalent statement that is easier to say and easier to understand. We have simplified this phrase. To simplify means to say it more simply, but not to lose or distort the meaning.

In mathematical language, roughly the same thing happens. One and the same thing can be said, written differently. What does it mean to simplify an expression? This means that for the original expression there are many equivalent expressions, that is, those that mean the same thing. And from all this variety we must choose the simplest, in our opinion, or the most suitable for our further purposes.

For example, consider the numeric expression . It will be equivalent to .

It will also be equivalent to the first two: .

It turns out that we have simplified our expressions and found the shortest equivalent expression.

For numeric expressions, you always need to do everything and get the equivalent expression as a single number.

Let's look at an example of a literal expression . Obviously, it will be simpler.

When simplifying literal expressions, it is necessary to perform all possible actions.

Is it always necessary to simplify an expression? No, sometimes it will be more convenient for us to have an equivalent but longer entry.

Example: you need to subtract a number from a number.

It is possible to calculate, but if the first number were represented by its equivalent notation: , then the calculations would be instantaneous: .

That is, a simplified expression is not always beneficial for us for further calculations.

Nevertheless, very often we are faced with a task that just sounds like “simplify the expression.”

Simplify the expression: .

Solution

1) Perform the actions in the first and second brackets: .

2) Let's calculate the products: .

Obviously, the last expression has a simpler form than the initial one. We've simplified it.

In order to simplify the expression, it must be replaced with an equivalent (equal).

To determine the equivalent expression you need:

1) perform all possible actions,

2) use the properties of addition, subtraction, multiplication and division to simplify calculations.

Properties of addition and subtraction:

1. Commutative property of addition: rearranging the terms does not change the sum.

2. Combinative property of addition: in order to add a third number to the sum of two numbers, you can add the sum of the second and third numbers to the first number.

3. The property of subtracting a sum from a number: to subtract a sum from a number, you can subtract each term separately.

Properties of multiplication and division

1. Commutative property of multiplication: rearranging the factors does not change the product.

2. Combinative property: to multiply a number by the product of two numbers, you can first multiply it by the first factor, and then multiply the resulting product by the second factor.

3. Distributive property of multiplication: in order to multiply a number by a sum, you need to multiply it by each term separately.

Let's see how we actually do mental calculations.

Calculate:

Solution

1) Let's imagine how

2) Let's imagine the first factor as a sum of bit terms and perform the multiplication:

3) you can imagine how and perform multiplication:

4) Replace the first factor with an equivalent sum:

The distributive law can also be used in reverse side: .

Follow these steps:

1) 2)

Solution

1) For convenience, you can use the distributive law, but use it in the opposite direction - take the common factor out of brackets.

2) Let’s take the common factor out of brackets

It is necessary to buy linoleum for the kitchen and hallway. Kitchen area - , hallway - . There are three types of linoleums: for, and rubles for. How much will each cost? three types linoleum? (Fig. 1)

Rice. 1. Illustration for the problem statement

Solution

Method 1. You can separately find out how much money it will take to buy linoleum for the kitchen, and then put it in the hallway and add up the resulting products.

An algebraic expression in which, along with the operations of addition, subtraction and multiplication, also uses division into letter expressions, is called a fractional algebraic expression. These are, for example, the expressions

We call an algebraic fraction algebraic expression, which has the form of the quotient of the division of two integer algebraic expressions (for example, monomials or polynomials). These are, for example, the expressions

The third of the expressions).

Identical transformations of fractional algebraic expressions are mostly aimed at representing them in the form of an algebraic fraction. To find the common denominator, factorization of the denominators of fractions is used - terms in order to find their least common multiple. When reducing algebraic fractions, the strict identity of expressions may be violated: it is necessary to exclude values ​​of quantities at which the factor by which the reduction is made becomes zero.

Let's give examples identity transformations fractional algebraic expressions.

Example 1: Simplify an expression

All terms can be reduced to a common denominator (it is convenient to change the sign in the denominator of the last term and the sign in front of it):

Our expression is equal to one for all values ​​except these values; it is undefined and reducing the fraction is illegal).

Example 2. Represent the expression as an algebraic fraction

Solution. The expression can be taken as a common denominator. We find sequentially:

Exercises

1. Find the values ​​of algebraic expressions for the specified parameter values:

2. Factorize.

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 terms of the polynomial. Monomials are also classified as polynomials, considering a monomial to be a polynomial consisting of one member.

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

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

Let us present 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 terms 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 degree of polynomial of a standard form take the highest of the powers of its members. Thus, the binomial \(12a^2b - 7b\) has the third degree, and the trinomial \(2b^2 -7b + 6\) has the second.

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

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

Sometimes the terms of a polynomial need to be divided into groups, enclosing each group in parentheses. Since enclosing parentheses is the inverse transformation of opening parentheses, it is easy to formulate rules for opening brackets:

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

If a “-” sign is placed before the brackets, then the terms enclosed in the brackets are written with opposite signs.

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

Using the distributive property of multiplication, you 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, you must multiply that monomial by each of the terms of the polynomial.

We have already used this rule several times to multiply by a sum.

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 the following rule is used.

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 squares, differences and difference of squares

With some expressions in algebraic transformations have to deal with more often than others. Perhaps the most common expressions are \((a + b)^2, \; (a - b)^2 \) and \(a^2 - b^2 \), i.e. the square of the sum, the square of the difference and difference of squares. You noticed that the names of these expressions seem to be incomplete, 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 does not occur very often; as a rule, instead of the letters a and b, it contains various, sometimes quite complex, expressions.

The expressions \((a + b)^2, \; (a - b)^2 \) can be easily converted (simplified) into polynomials of the standard form; in fact, you have already encountered this task when multiplying polynomials:
\((a + b)^2 = (a + b)(a + b) = a^2 + ab + ba + b^2 = \)
\(= a^2 + 2ab + b^2 \)

It is useful to remember the resulting identities and apply them without intermediate calculations. Brief verbal formulations help this.

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

\((a - b)^2 = a^2 + b^2 - 2ab \) - the square of the difference is equal to the sum of squares without the doubled 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 one to replace its left-hand parts with right-hand ones in transformations and vice versa - right-hand parts with left-hand ones. The most difficult thing is to see the corresponding expressions and understand how the variables a and b are replaced in them. Let's look at several examples of using abbreviated multiplication formulas.

Simplifying algebraic expressions is one of the key points 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 skills of simplification are good even for those who are not enthusiastic about mathematics. By observing several simple rules, you can simplify many of the most common types of algebraic expressions 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, similar terms include the same variable to the same degree, include several of the same 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 similar terms because they contain a second-order (to the second power) variable "x". However, x and x2 are not similar terms, since they contain the variable “x” of different orders (first and second). Likewise, -3yx and 5xz are not similar terms because they contain different variables.

2. Factorization. This is finding numbers whose product leads to the original number. Any original number can have several factors. For example, the number 12 can be decomposed into next row 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. Factors are the same as divisors, that is, the numbers by which the original number is divided .

   • 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

   Bringing similar members

   1. Write down the expression. Simple algebraic expressions (those that don't contain fractions, roots, etc.) can be solved (simplified) in just a few steps.

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

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

      • 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 terms (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 terms. 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 one.

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

   5. Follow the order of operations when bringing similar members. In our example, it was easy to provide similar terms. However, in the case of complex expressions in which terms are enclosed in parentheses 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 similar terms and present them, because it is necessary to open the parentheses first. Therefore, perform the operations according to 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 bring similar terms.
        • x 2 + (15x - 3x) + (8 - 5)
        • x 2 + 12x + 3

   Taking the multiplier out of brackets

   1. Find the greatest common divisor (GCD) of all the coefficients of the expression. GCD is greatest number, by which all coefficients of the expression are divided.

      • 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 term in the expression by 3.
        • 9x 2 /3 = 3x 2
        • 27x/3 = 9x
        • -3/3 = -1
        • The result was an expression 3x 2 + 9x - 1. It is not equal to the original expression.

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

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

   4. Simplifying fractional expressions by putting the factor out of brackets. Why simply put 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 brackets can help get rid of the fraction (from the denominator).

      • For example, consider fractional expression(9x 2 + 27x - 3)/3. Use factoring out to simplify this expression.
        • Put the factor of 3 out of brackets (as you did earlier): (3(3x 2 + 9x - 1))/3
        • Notice that there is now a 3 in both the numerator and the denominator. This can be reduced to give the expression: (3x 2 + 9x – 1)/1
        • Since any fraction that has the number 1 in the denominator is simply equal to the numerator, the original fraction expression simplifies to: 3x 2 + 9x - 1.

   Additional simplification methods

4. Let's look at a simple example: √(90). The number 90 can be factored into the following factors: 9 and 10, and extracted from 9 Square root(3) and remove 3 from under the root.
   • √(90)
   • √(9×10)
   • √(9)×√(10)
   • 3×√(10)
   • 3√(10)

5. Simplifying expressions with powers. Some expressions contain operations of multiplication or division of terms with powers. In the case of multiplying terms with the same base, their powers are added; in the case of dividing terms with the same base, their powers are subtracted.

   • For example, consider the expression 6x 3 × 8x 4 + (x 17 /x 15). In the case of multiplication, add the powers, and in the case of division, subtract them.
     • 6x 3 × 8x 4 + (x 17 /x 15)
     • (6 × 8)x 3 + 4 + (x 17 - 15)
     • 48x 7 + x 2
   • The following is an explanation of the rules for multiplying and dividing terms with powers.
     • 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 .
     • Likewise, dividing terms with degrees is equivalent to dividing terms by themselves. x 5 / x 3 = (x × x × x × x × x)/(x × x × x). Since similar terms found in both the numerator and the denominator can be reduced, the product of two “x”, or x 2 , remains in the numerator.

• Always remember about the signs (plus or minus) before the terms of the expression, as many people have difficulty choosing the correct sign.
• Ask for help if needed!
• Simplifying algebraic expressions isn't easy, but once you get the hang of it, it's a skill you can use for the rest of your life.

It is known that in mathematics there is no way to do without simplifying expressions. This is necessary for correctly and quickly solving a wide variety of problems, as well as various types of equations. The simplification discussed here implies a reduction in the number of actions required to achieve a goal. As a result, calculations are noticeably simplified and time is saved significantly. But how to simplify the expression? For this, established mathematical relationships are used, often called formulas, or laws, which allow expressions to be made much shorter, thereby simplifying calculations.

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

However, this is not possible with every expression. Therefore, let's take a closer look at more traditional methods.

Taking out the common divisor

In the case when one expression contains monomials that have the same factors, you can find the sum of their coefficients and then multiply by the common factor for them. This operation is also called "removal" common divisor". Consistently using this method, sometimes you can significantly simplify the expression. Algebra, in general, is built on the grouping and rearrangement of factors and divisors.

The simplest formulas for abbreviated multiplication

One of the consequences of the previously described method is the abbreviated multiplication formulas. How to simplify expressions with their help is much clearer to those who have not even memorized 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 mechanical and mathematical faculties. Difference of squares, square of difference and sum, 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 problems. Examples of such transformations can be easily found in any school algebra textbook, or, even easier, on the World Wide Web.

Degree roots

Elementary mathematics, if you look at it as a whole, does not have many ways to simplify an expression. Degrees and operations with them, as a rule, are relatively easy for most students. But many modern schoolchildren and students have considerable difficulties when it is necessary to simplify an expression with roots. And this is completely unfounded. Because mathematical nature roots are 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 more than the same number, variable or expression to the power of one-half, cube root– the same thing to the degree of “one third” and so on according to correspondence.

Simplifying expressions with fractions

Let's also look at a common example of how to simplify an expression with fractions. In cases where the expressions are natural fractions, you should isolate the common factor from the denominator and numerator, and then reduce the fraction by it. When monomials have identical factors raised to powers, it is necessary to ensure that the powers are equal when summing them up.

Simplifying basic trigonometric expressions

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

To summarize, we will provide the reader with some general advice:

• Polynomials should be factorized, that is, they should be represented in the form of a product of a certain number of factors - monomials and polynomials. If such a possibility exists, it is necessary to take the common factor out of brackets.
• It’s better to memorize all abbreviated multiplication formulas without exception. There are not so many of them, but they are the basis for simplifying mathematical expressions. We should also not forget about the method of isolating perfect squares in trinomials, which is the inverse action to one of the abbreviated multiplication formulas.
• All fractions present in the expression should be reduced as often as possible. However, do not forget that only the multipliers are reduced. When the denominator and numerator of algebraic fractions are multiplied by the same number, which is different from zero, the meanings of the fractions do not change.
• In general, all expressions can be transformed by actions, or in a chain. The first method is more preferable, because the results of intermediate actions are easier to verify.
• Quite often in mathematical expressions we have to extract roots. It should be remembered that the roots of even powers can be extracted only from a non-negative number or expression, and the roots of odd powers can be extracted from absolutely any expressions or numbers.

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