Rootnth degree and its main properties
Degree real number a with a natural indicator P there is a work P factors, each of which is equal to a:
a1 = a; a2 = a a; a n =
For example,
25 = 2 2 2 2 2 = 32,
Five times
(-3)4 = (-3)(-3)(-3)(-3) = 81.
4 times
Real number a called the basis of the degree and a natural number n - degree indicator.
The main properties of degrees with natural exponents follow directly from the definition: the degree of a positive number with any P e N positive; the degree of a negative number with an even exponent is positive, with an odd number it is negative.
For example,
(-5)4 = (-5) (-5) (-5) (-5) = 625; (-5)3 = (-5)-(-5)-(-5) = -125.
Actions with degrees are performed according to the following rules.
1. To multiply powers with the same base, it is enough to add the exponents, and leave the base the same, that is
For example, p5∙p3 = p5+3 =p8
2. To divide degrees with the same bases, it is enough to subtract the divisor indicator from the dividend indicator, and leave the base the same, that is
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2. To raise a power to a power, it is enough to multiply the exponents, leaving the base the same, that is
(ap)m = at p. For example, (23)2 = 26.
4. To raise a product to a power, it is enough to raise each factor to this power and multiply the results, that is
(a b)P= an ∙bP.
For example, (2y3)2= 4y6.
5. To raise a fraction to a power, it is enough to raise the numerator and denominator separately to this power and divide the first result by the second, that is
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Note that these formulas are sometimes useful to read from right to left. In this case, they become rules. For example, in case 4, apvp= (av)p we get the following rule: to to multiply powers with the same exponents, it is enough to multiply the bases, leaving the exponent the same.
Using this rule is effective, for example, when calculating the following product
(https://pandia.ru/text/78/410/images/image006_27.gif" width="25" height="23">+1)5=(( -1)( +1))5=( = 1.
We now give the definition of a root.
root nth degree from a real number a called a real number X, whose nth power is a.
Obviously, in accordance with the basic properties of degrees with natural exponents, from any positive number there are two opposite values of the root of an even degree, for example, the numbers 4 and -4 are the square roots of 16, since (-4) 2 \u003d 42 \u003d 16, and the numbers 3 and -3 are the fourth roots of 81, since (-3)4 = 34 = 81.
Also, there is no even root of a negative number, because an even power of any real number is non-negative. As for the root of an odd degree, then for any real number there is only one root of an odd degree from this number. For example, 3 is the third root of 27 because Z3 = 27, and -2 is the fifth root of -32 because (-2)5 = 32.
In connection with the existence of two roots of an even degree from a positive number, we introduce the concept of an arithmetic root in order to eliminate this ambiguity of the root.
The non-negative value of the n-th root of a non-negative number is called arithmetic root.
For example, https://pandia.ru/text/78/410/images/image008_21.gif" width="13" height="16 src="> 0.
It should be remembered that when solving irrational equations, their roots are always considered as arithmetic.
We note the main property of the root of the nth degree.
The value of the root will not change if the indicators of the root and the degree of the root expression are multiplied or divided by the same natural number, that is
Example 7. Reduce to a common denominator and
The material of this article should be considered as part of the topic transformation of irrational expressions. Here, using examples, we will analyze all the subtleties and nuances (of which there are many) that arise when carrying out transformations based on the properties of the roots.
Page navigation.
Recall the properties of roots
Since we are going to deal with the transformation of expressions using the properties of the roots, it does not hurt to remember the main ones, or even better, write them down on paper and place them in front of you.
First, square roots and their following properties are studied (a, b, a 1, a 2, ..., a k - real numbers):
And later, the idea of the root is expanded, the definition of the root of the nth degree is introduced, and such properties are considered (a, b, a 1, a 2, ..., a k are real numbers, m, n, n 1, n 2, ... , n k - natural numbers):
Converting expressions with numbers under root signs
As usual, they first learn to work with numerical expressions, and after that they move on to expressions with variables. We will do the same, and first we will deal with the transformation irrational expressions, containing under signs of roots only numeric expressions, and already further in the next paragraph we will introduce under the signs of the roots and variables.
How can this be used to transform expressions? Very simple: for example, we can replace an irrational expression with an expression, or vice versa. That is, if the converted expression contains an expression that matches the expression from the left (right) part of any of the listed properties of the roots, then it can be replaced by the corresponding expression from the right (left) part. This is the transformation of expressions using the properties of the roots.
Let's take a few more examples.
Let's simplify the expression 
. The numbers 3 , 5 and 7 are positive, so we can safely apply the properties of the roots. Here you can act differently. For example, a property-based root can be represented as , and a property-based root with k=3 as , with this approach, the solution will look like this: 
It was possible to do otherwise, replacing with , and then with , in this case the solution would look like this: 
Other solutions are possible, for example:
Let's take a look at another example. Let's transform the expression. Looking at the list of properties of the roots, we select from it the properties we need to solve the example, it is clear that two of them and are useful here, which are valid for any a . We have: 
Alternatively, one could first transform expressions under root signs using 
and then apply the properties of the roots
Up to this point, we have converted expressions that contain only square roots. It's time to work with roots that have other indicators.
Example.
Transform Irrational Expression 
.
Solution.
By property 
the first factor of a given product can be replaced by the number −2: 
Move on. The second factor, by virtue of the property, can be represented as, and it does not hurt to replace 81 with the quadruple power of three, since the number 3 appears in the remaining factors under the signs of the roots: 
It is advisable to replace the fraction root with the ratio of the roots of the form , which can be further transformed: 
. We have
The resulting expression after performing operations with twos will take the form , and it remains to transform the product of the roots.
To transform the products of the roots, they are usually reduced to one indicator, for which it is advisable to take the indicators of all roots. In our case, LCM(12, 6, 12)=12 , and only the root will have to be reduced to this indicator, since the other two roots already have such an indicator. To cope with this task allows equality, which is applied from right to left. So . Considering this result, we have 
Now the product of the roots can be replaced by the root of the product and the remaining, already obvious, transformations can be performed: 
Let's make a short version of the solution: 
Answer:
.
Separately, we emphasize that in order to apply the properties of the roots, it is necessary to take into account the restrictions imposed on the numbers under the signs of the roots (a≥0, etc.). Ignoring them can lead to incorrect results. For example, we know that the property holds for non-negative a . Based on it, we can safely go, for example, from to, since 8 is a positive number. But if we take a meaningful root of a negative number, for example, , and, based on the above property, replace it with , then we will actually replace −2 with 2 . Indeed, , a . That is, for negative a, the equality may be false, just as other properties of the roots may be false without taking into account the conditions specified for them.
But what was said in the previous paragraph does not mean at all that expressions with negative numbers under the root signs cannot be transformed using the properties of the roots. They just need to be “prepared” beforehand by applying the rules of operations with numbers or using the definition of the root of an odd degree from a negative number, which corresponds to the equality , where −a is a negative number (while a is positive). For example, it cannot be immediately replaced by , since −2 and −3 are negative numbers, but it allows us to move from the root to , and then apply the property of the root from the product: 
. And in one of the previous examples, it was necessary to move from the root to the root of the eighteenth degree not like this, but like this 
.
So, to transform expressions using the properties of the roots, you need to
- select the appropriate property from the list,
- make sure that the numbers under the root satisfy the conditions for the selected property (otherwise, you need to perform preliminary transformations),
- and carry out the intended transformation.
Converting expressions with variables under root signs
To transform irrational expressions containing not only numbers but also variables under the root sign, the properties of the roots listed in the first paragraph of this article must be applied carefully. This is due for the most part to the conditions that must be satisfied by the numbers involved in the formulas. For example, based on the formula , the expression can be replaced by an expression only for those x values that satisfy the conditions x≥0 and x+1≥0 , since the indicated formula is set for a≥0 and b≥0 .
What is the danger of ignoring these conditions? The answer to this question is clearly demonstrated by the following example. Let's say we need to calculate the value of an expression when x=−2 . If we immediately substitute the number −2 instead of the variable x, then we get the value we need 
. And now let's imagine that, based on some considerations, we converted the given expression to the form , and only after that we decided to calculate the value. We substitute the number −2 instead of x and arrive at the expression 
, which doesn't make sense.
Let's see what happens to the range of valid values (ODV) of the x variable as we move from expression to expression. We mentioned the ODZ not by chance, since this is a serious tool for controlling the admissibility of the transformations performed, and changing the ODZ after the transformation of the expression should at least alert. It is not difficult to find the ODZ for these expressions. For the expression, the ODZ is determined from the inequality x (x+1)≥0 , its solution gives the numerical set (−∞, −1]∪∪∪)