When we get acquainted with the root in school, we study the concept of irrational expressions. Such expressions are closely related to the roots.
Irrational expressions Are expressions that have a root. That is, these are expressions that have radicals.
Based on this definition, we have that x - 1, 8 3 · 3 6 - 1 2 · 3, 7 - 4 · 3 · (2 + 3), 4 · a 2 d 5: d 9 2 · a 3 5 - these are all expressions of an irrational type.
When considering the expression x · x - 7 · x + 7 x + 3 2 · x - 8 3 we get that the expression is rational. Rational expressions include polynomials and algebraic fractions. Irrational ones include working with logarithmic expressions or rooted expressions.
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At the beginning of the lesson, we will repeat the basic properties of square roots, and then consider some complex examples to simplify expressions containing square roots.
If you have difficulty understanding the topic, we recommend that you look at the lesson “Simplification of Expressions”
Repeating square root properties
We briefly repeat the theory and recall the basic properties of square roots.
Properties of square roots:
Examples for simplifying expressions with roots
Let's move on to examples of using these properties.
Example 1 Simplify the expression.
Decision. To simplify, the number 120 must be decomposed into simple factors:
. We will reveal the square of the sum according to the corresponding formula:
Example 2 Simplify the expression.
Decision. We will take into account that this expression does not make sense for all possible values of the variable, since the square roots and fractions are present in this expression, which leads to a “narrowing” of the range of permissible values. ODZ:).
We give the expression in parentheses to the common denominator and write the numerator of the last fraction as the difference of squares:
Example 3 Simplify the expression.
Decision. It can be seen that the second bracket of the numerator has an uncomfortable appearance and needs to be simplified, we will try to factor it using the grouping method.
. To be able to factor out a common factor, we simplified the roots by factoring them. Substitute the resulting expression in the original fraction:
. After reducing the fraction, we apply the formula of the difference of squares.
An example of getting rid of irrationality
Example 4 To get rid of irrationality (roots) in the denominator: a).
Decision. a) In order to get rid of irrationality in the denominator, the standard method of multiplying both the numerator and denominator of the fraction by the factor conjugate to the denominator is used (the same expression, but with the opposite sign). This is done to supplement the denominator of the fraction to the difference of the squares, which allows you to get rid of the roots in the denominator. We perform this technique in our case:
b) we perform similar actions:
An example of the proof and the allocation of a full square in a complex radical
Example 5 Prove the equality.
Evidence. We use the definition of the square root, from which it follows that the square of the right expression must be equal to the root expression:
. We will reveal the brackets according to the formula of the square of the sum:
, got true equality.
Example 6 Simplify the expression.
Decision. The indicated expression is usually called a complex radical (root under the root). In this example, you must guess to select the full square from the root expression. For this, we note that of the two terms, and for the role of the second - 1.
. Substitute this expression at the root:
In this lesson, we end the topic “Function. The properties of the square root ”, and in the next lesson we begin the new topic“ Real numbers ”.
1. Bashmakov M.I. Algebra grade 8. - M.: Education, 2004.
2. Dorofeev G.V., Suvorova SB, Bunimovich EA and other Algebra 8. - 5th ed. - M .: Education, 2010.
3. Nikolsky S. M., Potapov M. A., Reshetnikov N. N., Shevkin A. V. Algebra grade 8. Textbook for educational institutions. - M.: Education, 2006.
Additional recommended links to Internet resources
1. The Internet portal xenoid.ru (Source).
2. Mathematical school (Source).
3. Internet portal XReferat.Ru (Source).
1. No. 357, 360, 372, 373, 382. Dorofeev G.V., Suvorova SB, Bunimovich EA and other Algebra 8. - 5th ed. - M .: Education, 2010.
2. Get rid of irrationality in the denominator: a).
3. Simplify the expression: a).
4. Prove the identity.
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The main types of transformations of irrational expressions
When calculating such expressions, it is necessary to pay attention to the DLD. Often they require additional transformations in the form of opening brackets, casting similar members, groups, and so on. The basis of such transformations is actions with numbers. Transformations of irrational expressions adhere to a strict order.
Convert the expression 9 + 3 3 - 2 + 4 · 3 3 + 1 - 2 · 3 3.
You must replace the number 9 with an expression containing the root. Then we get that
81 + 3 3 - 2 + 4 · 3 3 + 1 - 2 · 3 3 = = 9 + 3 3 - 2 + 4 · 3 3 + 1 - 2 · 3 3
The resulting expression has similar terms; therefore, we perform the reduction and grouping. Get
9 + 3 3 - 2 + 4 · 3 3 + 1 - 2 · 3 3 = = 9 - 2 + 1 + 3 3 + 4 · 3 3 - 2 · 3 3 = = 8 + 3 · 3 3
Answer: 9 + 3 3 - 2 + 4 · 3 3 + 1 - 2 · 3 3 = 8 + 3 · 3 3
Present the expression x + 3 5 2 - 2 · x + 3 5 + 1 - 9 in the form of a product of two irrational ones using the abbreviated multiplication formulas.
x + 3 5 2 - 2 x + 3 5 + 1 - 9 = = x + 3 5 - 1 2 - 9
We represent 9 in the form of 3 2, and we apply the formula of the difference of squares:
x + 3 5 - 1 2 - 9 = x + 3 5 - 1 2 - 3 2 = = x + 3 5 - 1 - 3 · x + 3 5 - 1 + 3 = = x + 3 5 - 4 · x + 3 5 + 2
The result of the identical transformations led to the product of two rational expressions that needed to be found.
x + 3 5 2 - 2 x + 3 5 + 1 - 9 = = x + 3 5 - 4 x + 3 5 + 2
You can perform a number of other transformations that relate to irrational expressions.
Convert root expression
It is important that the expression under the sign of the root can be replaced by an identically equal to it. This statement makes it possible to work with the radical expression. For example, 1 + 6 can be replaced by 7 or 2 · a 5 4 - 6 by 2 · a 4 · a 4 - 6. They are identically equal, therefore replacement makes sense.
When there is no a 1 other than a, where an inequality of the form a n = a 1 n is true, then such an equality is possible only for a = a 1. The values of such expressions are equal with any values of the variables.
Using root properties
Root properties are used to simplify expressions. To apply the property a · b = a · b, where a ≥ 0, b ≥ 0, then from the irrational form 1 + 3 · 12 we can become identically equal to 1 + 3 · 12. Property. . . a n k n 2 n 1 = a n 1 · n 2 ·,. . . , · N k, where a ≥ 0 indicates that x 2 + 4 4 3 can be written in the form x 2 + 4 24.
There are some nuances in the transformation of radical expressions. If there is an expression, then - 7 - 81 4 = - 7 4 - 81 4 we cannot write, since the formula a b n = a n b n serves only for non-negative a and positive b. If the property is applied correctly, then we get an expression of the form 7 4 81 4.
For the correct transformation, transformations of irrational expressions using the properties of the roots are used.
Introducing a multiplier under the root sign
Enter under the root sign - means to replace the expression B · C n, and B and C are some numbers or expressions, where n is a natural number that is greater than 1, an equal expression that has the form B n · C n or - B n · C n.
If we simplify the expression of the form 2 · x 3, then after entering under the root, we get that 2 3 · x 3. Such transformations are possible only after a detailed study of the rules for introducing a factor under the root sign.
Extraction of the factor from under the root sign
If there is an expression of the form B n · C n, then it is reduced to the form B · C n, where there are odd n, which take the form B · C n with even n, B and C are some numbers and expressions.
That is, if we take an irrational expression of the form 2 3 · x 3, take out the factor from under the root, then we get the expression 2 · x 3. Or x + 1 2 · 7 will result in an expression of the form x + 1 · 7, which has another notation in the form x + 1 · 7.
Pulling the factor out from under the root is necessary to simplify the expression and transform it quickly.
Convert fractions containing roots
An irrational expression can be either a natural number or a fraction. To convert fractional expressions, much attention is paid to its denominator. If we take a fraction of the form (2 + 3) · x 4 x 2 + 5 3, then the numerator will take the form 5 · x 4, and using the properties of the roots, we get that the denominator becomes x 2 + 5 6. The initial fraction can be written in the form 5 · x 4 x 2 + 5 6.
It is necessary to pay attention to the fact that it is necessary to change the sign of only the numerator or only the denominator. We get that
- x + 2 · x - 3 · x 2 + 7 4 = x + 2 · x - (- 3 · x 2 + 7 4) = x + 2 · x 3 · x 2 - 7 4
Fraction reduction is most often used in simplification. We get that
3 · x + 4 3 - 1; x x + 4 3 - 1 3 we reduce by x + 4 3 - 1. We get the expression 3 · x x + 4 3 - 1 2.
Before reduction, it is necessary to perform transformations that simplify the expression and make it possible to factor the complex expression. The most commonly used formulas are abbreviated multiplication.
If we take a fraction of the form 2 · x - y x + y, then it is necessary to introduce new variables u = x and v = x, then the given expression will change the form and become 2 · u 2 - v 2 u + v. The numerator should be decomposed into polynomials by the formula, then we get that
2 · u 2 - v 2 u + v = 2 · (u - v) · u + v u + v = 2 · u - v. After performing the reverse replacement, we come to the form 2 · x - y, which is equal to the original one.
Reduction to a new denominator is allowed, then it is necessary to multiply the numerator by an additional factor. If we take a fraction of the form x 3 - 1 0, 5 · x, then we bring to the denominator x. for this you need to multiply the numerator and denominator by the expression 2 · x, then we get the expression x 3 - 1 0, 5 · x = 2 · x · x 3 - 1 0, 5 · x · 2 · x = 2 · x · x 3 - 1 x.
Reduction of fractions or reduction of similar fractions is necessary only on the ODZ of the indicated fraction. When we multiply the numerator and the denominator by the irrational expression, we get that we get rid of the irrationality in the denominator.
Transition from roots to degrees
Transitions from roots to degrees are necessary for the quick conversion of irrational expressions. If we consider the equality a m n = a m n, then we can see that its use is possible when a is a positive number, m is an integer, and n is a natural number. If we consider the expression 5 - 2 3, then otherwise we have the right to write it as 5 - 2 3. These expressions are equivalent.
When there is a negative number or a number with variables under the root, then the formula a m n = a m n is not always applicable. If it is necessary to replace such roots with (- 8) 3 5 and (- 16) 2 4 degrees, then we get that - 8 3 5 and - 16 2 4 by the formula a m n = a m n do not work with negative a. in order to analyze in detail the topic of radical expressions and their simplifications, it is necessary to study the article on the transition from roots to degrees and vice versa. It should be remembered that the formula a m n = a m n is not applicable to all expressions of this kind. Getting rid of irrationality contributes to the further simplification of expression, its transformation and solution.