Rootn-th degree and its basic properties
Degree real number A with natural indicator P there is a work P factors, each of which is equal A:
a1 = a; a2 = a · a; A n =
For example,
25 = 2 2 2 2 2 = 32,
5 times
(-3)4 = (-3)(-3)(-3)(-3) = 81.
4 times
Real number A called the basis of the degree, and the natural number n is exponent.
The basic properties of powers with natural exponents follow directly from the definition: power of a positive number with any P e N positive; The power of a negative number with an even exponent is positive, with an odd exponent it is negative.
For example,
(-5)4 = (-5) (-5) (-5) (-5) = 625; (-5)3 = (-5)-(-5)-(-5) = -125.
Actions with degrees are performed as follows: rules.
1. To multiply powers with the same bases, 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 powers with the same bases, it is enough to subtract the exponent of the divisor from the index of the dividend and leave the base the same, that is
https://pandia.ru/text/78/410/images/image003_63.gif" width="95" height="44 src=">
2. To raise a degree 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= ap∙bP.
For example, (2у3)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
https://pandia.ru/text/78/410/images/image005_37.gif" width="87" height="53 src=">
Note that it is sometimes useful to read these formulas 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.
Let us now give the definition of a root.
Root nth degree from a real number A called a real number X, the nth power of which is equal to A.
Obviously, in accordance with the basic properties of powers with natural exponents, from any positive number there are two opposite values of the root of an even power, for example, the numbers 4 and -4 are square roots of 16, since (-4)2 = 42 = 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 the even power of any real number is non-negative. As for the odd root, for any real number there is only one odd root of that number. For example, 3 is the third root of 27, since 33 = 27, and -2 is the fifth root of -32, since (-2)5 = 32.
Due to the existence of two even-degree roots of a positive number, we introduce the concept of an arithmetic root to eliminate this ambiguity of the root.
The non-negative value of the nth 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.
Let us note the main property of the nth root.
The size of the root will not change if the indicators of the root and the degree of the radical expression are multiplied or divided by the same natural number, that is
Example 7. Reduce to a common denominator and
Greetings, cats! Last time we discussed in detail what roots are (if you don’t remember, I recommend reading it). The main takeaway from that lesson: there is only one universal definition of roots, which is what you need to know. The rest is nonsense and a waste of time.
Today we go further. We will learn to multiply roots, we will study some problems associated with multiplication (if these problems are not solved, they can become fatal in the exam) and we will practice properly. So stock up on popcorn, get comfortable, and let's get started. :)
You haven't smoked it yet either, have you?
The lesson turned out to be quite long, so I divided it into two parts:
- First we will look at the rules of multiplication. Cap seems to be hinting: this is when there are two roots, between them there is a “multiply” sign - and we want to do something with it.
- Then let's look at the opposite situation: there is one big root, but we were eager to represent it as a product of two simpler roots. Why is this necessary, is a separate question. We will only analyze the algorithm.
For those who can’t wait to immediately move on to the second part, you are welcome. Let's start with the rest in order.
Basic Rule of Multiplication
Let's start with the simplest thing - classic square roots. The same ones that are denoted by $\sqrt(a)$ and $\sqrt(b)$. Everything is obvious to them:
Multiplication rule. To multiply one square root by another, you simply multiply their radical expressions, and write the result under the common radical:
\[\sqrt(a)\cdot \sqrt(b)=\sqrt(a\cdot b)\]
No additional restrictions are imposed on the numbers on the right or left: if the root factors exist, then the product also exists.
Examples. Let's look at four examples with numbers at once:
\[\begin(align) & \sqrt(25)\cdot \sqrt(4)=\sqrt(25\cdot 4)=\sqrt(100)=10; \\ & \sqrt(32)\cdot \sqrt(2)=\sqrt(32\cdot 2)=\sqrt(64)=8; \\ & \sqrt(54)\cdot \sqrt(6)=\sqrt(54\cdot 6)=\sqrt(324)=18; \\ & \sqrt(\frac(3)(17))\cdot \sqrt(\frac(17)(27))=\sqrt(\frac(3)(17)\cdot \frac(17)(27 ))=\sqrt(\frac(1)(9))=\frac(1)(3). \\ \end(align)\]
As you can see, the main meaning of this rule is to simplify irrational expressions. And if in the first example we ourselves would have extracted the roots of 25 and 4 without any new rules, then things get tough: $\sqrt(32)$ and $\sqrt(2)$ are not considered by themselves, but their product turns out to be a perfect square, so its root is equal to a rational number.
I would especially like to highlight the last line. There, both radical expressions are fractions. Thanks to the product, many factors are canceled, and the entire expression turns into an adequate number.
Of course, things won't always be so beautiful. Sometimes there will be a complete mess under the roots - it is not clear what to do with it and how to transform it after multiplication. A little later, when you start studying irrational equations and inequalities, there will be all sorts of variables and functions. And very often, problem writers count on the fact that you will discover some canceling terms or factors, after which the problem will be simplified many times over.
In addition, it is not at all necessary to multiply exactly two roots. You can multiply three, four, or even ten at once! This will not change the rule. Take a look:
\[\begin(align) & \sqrt(2)\cdot \sqrt(3)\cdot \sqrt(6)=\sqrt(2\cdot 3\cdot 6)=\sqrt(36)=6; \\ & \sqrt(5)\cdot \sqrt(2)\cdot \sqrt(0.001)=\sqrt(5\cdot 2\cdot 0.001)= \\ & =\sqrt(10\cdot \frac(1) (1000))=\sqrt(\frac(1)(100))=\frac(1)(10). \\ \end(align)\]
And again a small note on the second example. As you can see, in the third factor under the root there is a decimal fraction - in the process of calculations we replace it with a regular one, after which everything is easily reduced. So: I highly recommend getting rid of decimal fractions in any irrational expressions (i.e. containing at least one radical symbol). This will save you a lot of time and nerves in the future.
But this was a lyrical digression. Now let's consider a more general case - when the root exponent contains an arbitrary number $n$, and not just the “classical” two.
The case of an arbitrary indicator
So, we've sorted out the square roots. What to do with cubic ones? Or even with roots of arbitrary degree $n$? Yes, everything is the same. The rule remains the same:
To multiply two roots of degree $n$, it is enough to multiply their radical expressions, and then write the result under one radical.
In general, nothing complicated. Except that the amount of calculations may be greater. Let's look at a couple of examples:
Examples. Calculate products:
\[\begin(align) & \sqrt(20)\cdot \sqrt(\frac(125)(4))=\sqrt(20\cdot \frac(125)(4))=\sqrt(625)= 5; \\ & \sqrt(\frac(16)(625))\cdot \sqrt(0.16)=\sqrt(\frac(16)(625)\cdot \frac(16)(100))=\sqrt (\frac(64)(((25)^(2))\cdot 25))= \\ & =\sqrt(\frac(((4)^(3)))(((25)^(3 ))))=\sqrt(((\left(\frac(4)(25) \right))^(3)))=\frac(4)(25). \\ \end(align)\]
And again, attention to the second expression. We multiply the cube roots, get rid of the decimal fraction and end up with the denominator being the product of the numbers 625 and 25. This is quite a large number - personally, I personally can’t figure out what it equals off the top of my head.
So we simply isolated the exact cube in the numerator and denominator, and then used one of the key properties (or, if you prefer, definition) of the $n$th root:
\[\begin(align) & \sqrt(((a)^(2n+1)))=a; \\ & \sqrt(((a)^(2n)))=\left| a\right|. \\ \end(align)\]
Such “machinations” can save you a lot of time on an exam or test, so remember:
Don't rush to multiply numbers using radical expressions. First, check: what if the exact degree of any expression is “encrypted” there?
Despite the obviousness of this remark, I must admit that most unprepared students do not see the exact degrees at point-blank range. Instead, they multiply everything outright, and then wonder: why did they get such brutal numbers? :)
However, all this is baby talk compared to what we will study now.
Multiplying roots with different exponents
Okay, now we can multiply roots with the same indicators. What if the indicators are different? Let's say, how to multiply an ordinary $\sqrt(2)$ by some crap like $\sqrt(23)$? Is it even possible to do this?
Yes of course you can. Everything is done according to this formula:
Rule for multiplying roots. To multiply $\sqrt[n](a)$ by $\sqrt[p](b)$, it is enough to perform the following transformation:
\[\sqrt[n](a)\cdot \sqrt[p](b)=\sqrt(((a)^(p))\cdot ((b)^(n)))\]
However, this formula only works if radical expressions are non-negative. This is a very important note that we will return to a little later.
For now, let's look at a couple of examples:
\[\begin(align) & \sqrt(3)\cdot \sqrt(2)=\sqrt(((3)^(4))\cdot ((2)^(3)))=\sqrt(81 \cdot 8)=\sqrt(648); \\ & \sqrt(2)\cdot \sqrt(7)=\sqrt(((2)^(5))\cdot ((7)^(2)))=\sqrt(32\cdot 49)= \sqrt(1568); \\ & \sqrt(5)\cdot \sqrt(3)=\sqrt(((5)^(4))\cdot ((3)^(2)))=\sqrt(625\cdot 9)= \sqrt(5625). \\ \end(align)\]
As you can see, nothing complicated. Now let's figure out where the non-negativity requirement came from, and what will happen if we violate it. :)
Multiplying roots is easy Why must radical expressions be non-negative?
Of course, you can be like school teachers and quote the textbook with a smart look:
The requirement of non-negativity is associated with different definitions of roots of even and odd degrees (accordingly, their domains of definition are also different).
Well, has it become clearer? Personally, when I read this nonsense in the 8th grade, I understood something like the following: “The requirement of non-negativity is associated with *#&^@(*#@^#)~%” - in short, I didn’t understand a damn thing at that time. :)
So now I’ll explain everything in a normal way.
First, let's find out where the multiplication formula above comes from. To do this, let me remind you of one important property of the root:
\[\sqrt[n](a)=\sqrt(((a)^(k)))\]
In other words, we can easily raise the radical expression to any natural power $k$ - in this case, the exponent of the root will have to be multiplied by the same power. Therefore, we can easily reduce any roots to overall indicator, then multiply. This is where the multiplication formula comes from:
\[\sqrt[n](a)\cdot \sqrt[p](b)=\sqrt(((a)^(p)))\cdot \sqrt(((b)^(n)))= \sqrt(((a)^(p))\cdot ((b)^(n)))\]
But there is one problem that sharply limits the use of all these formulas. Consider this number:
According to the formula just given, we can add any degree. Let's try adding $k=2$:
\[\sqrt(-5)=\sqrt(((\left(-5 \right))^(2)))=\sqrt(((5)^(2)))\]
We removed the minus precisely because the square burns the minus (like any other even degree). Now let’s perform the reverse transformation: “reduce” the two in the exponent and power. After all, any equality can be read both from left to right and from right to left:
\[\begin(align) & \sqrt[n](a)=\sqrt(((a)^(k)))\Rightarrow \sqrt(((a)^(k)))=\sqrt[n ](a); \\ & \sqrt(((a)^(k)))=\sqrt[n](a)\Rightarrow \sqrt(((5)^(2)))=\sqrt(((5)^( 2)))=\sqrt(5). \\ \end(align)\]
But then it turns out to be some kind of crap:
\[\sqrt(-5)=\sqrt(5)\]
This cannot happen, because $\sqrt(-5) \lt 0$, and $\sqrt(5) \gt 0$. This means that for even powers and negative numbers our formula no longer works. After which we have two options:
- To hit the wall and state that mathematics is a stupid science, where “there are some rules, but these are imprecise”;
- Introduce additional restrictions under which the formula will become 100% working.
In the first option, we will have to constantly catch “non-working” cases - it’s difficult, time-consuming and generally ugh. Therefore, mathematicians preferred the second option. :)
But don't worry! In practice, this limitation does not affect the calculations in any way, because all the problems described concern only roots of odd degree, and minuses can be taken from them.
Therefore, let us formulate one more rule, which generally applies to all actions with roots:
Before multiplying roots, make sure that the radical expressions are non-negative.
Example. In the number $\sqrt(-5)$ you can remove the minus from under the root sign - then everything will be normal:
\[\begin(align) & \sqrt(-5)=-\sqrt(5) \lt 0\Rightarrow \\ & \sqrt(-5)=-\sqrt(((5)^(2))) =-\sqrt(25)=-\sqrt(((5)^(2)))=-\sqrt(5) \lt 0 \\ \end(align)\]
Do you feel the difference? If you leave a minus under the root, then when the radical expression is squared, it will disappear, and crap will begin. And if you first take out the minus, then you can square/remove until you’re blue in the face - the number will remain negative. :)
Thus, the most correct and most reliable way to multiply roots is as follows:
- Remove all the negatives from the radicals. Minuses exist only in roots of odd multiplicity - they can be placed in front of the root and, if necessary, reduced (for example, if there are two of these minuses).
- Perform multiplication according to the rules discussed above in today's lesson. If the indicators of the roots are the same, we simply multiply the radical expressions. And if they are different, we use the evil formula \[\sqrt[n](a)\cdot \sqrt[p](b)=\sqrt(((a)^(p))\cdot ((b)^(n) ))\].
- 3.Enjoy the result and good grades.:)
Well? Shall we practice?
Example 1: Simplify the expression:
\[\begin(align) & \sqrt(48)\cdot \sqrt(-\frac(4)(3))=\sqrt(48)\cdot \left(-\sqrt(\frac(4)(3) )) \right)=-\sqrt(48)\cdot \sqrt(\frac(4)(3))= \\ & =-\sqrt(48\cdot \frac(4)(3))=-\ sqrt(64)=-4; \end(align)\]
This is the simplest option: the roots are the same and odd, the only problem is that the second factor is negative. We take this minus out of the picture, after which everything is easily calculated.
Example 2: Simplify the expression:
\[\begin(align) & \sqrt(32)\cdot \sqrt(4)=\sqrt(((2)^(5)))\cdot \sqrt(((2)^(2)))= \sqrt(((\left(((2)^(5)) \right))^(3))\cdot ((\left(((2)^(2)) \right))^(4) ))= \\ & =\sqrt(((2)^(15))\cdot ((2)^(8)))=\sqrt(((2)^(23))) \\ \end( align)\]
Here, many would be confused by the fact that the output turned out to be an irrational number. Yes, it happens: we couldn’t completely get rid of the root, but at least we significantly simplified the expression.
Example 3: Simplify the expression:
\[\begin(align) & \sqrt(a)\cdot \sqrt(((a)^(4)))=\sqrt(((a)^(3))\cdot ((\left((( a)^(4)) \right))^(6)))=\sqrt(((a)^(3))\cdot ((a)^(24)))= \\ & =\sqrt( ((a)^(27)))=\sqrt(((a)^(3\cdot 9)))=\sqrt(((a)^(3))) \end(align)\]
I would like to draw your attention to this task. There are two points here:
- The root is not a specific number or power, but the variable $a$. At first glance, this is a little unusual, but in reality, when solving mathematical problems, you most often have to deal with variables.
- In the end, we managed to “reduce” the radical indicator and the degree in radical expression. This happens quite often. And this means that it was possible to significantly simplify the calculations if you did not use the basic formula.
For example, you could do this:
\[\begin(align) & \sqrt(a)\cdot \sqrt(((a)^(4)))=\sqrt(a)\cdot \sqrt(((\left(((a)^( 4)) \right))^(2)))=\sqrt(a)\cdot \sqrt(((a)^(8))) \\ & =\sqrt(a\cdot ((a)^( 8)))=\sqrt(((a)^(9)))=\sqrt(((a)^(3\cdot 3)))=\sqrt(((a)^(3))) \ \\end(align)\]
In fact, all transformations were performed only with the second radical. And if you do not describe in detail all the intermediate steps, then in the end the amount of calculations will be significantly reduced.
In fact, we have already encountered a similar task above when we solved the example $\sqrt(5)\cdot \sqrt(3)$. Now it can be written much simpler:
\[\begin(align) & \sqrt(5)\cdot \sqrt(3)=\sqrt(((5)^(4))\cdot ((3)^(2)))=\sqrt(( (\left(((5)^(2))\cdot 3 \right))^(2)))= \\ & =\sqrt(((\left(75 \right))^(2))) =\sqrt(75). \end(align)\]
Well, we've sorted out the multiplication of roots. Now let's consider the reverse operation: what to do when there is a product under the root?
The material in this article should be considered as part of the topic transformation of irrational expressions. Here we will use examples to analyze all the subtleties and nuances (of which there are many) that arise when carrying out transformations based on the properties of roots.
Page navigation.
Let us recall the properties of roots
Since we are about to deal with the transformation of expressions using the properties of roots, it won’t 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 are real numbers):
And later the idea of a root is expanded, the definition of a root of the nth degree is introduced, and the following 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 radical signs
As usual, they first learn to work with numerical expressions, and only after that they move on to expressions with variables. We will do the same, and first we will deal with the transformation of irrational expressions containing only numerical expressions under the signs of the roots, and then in the next paragraph we will introduce variables under the signs of the roots.
How can this be used to transform expressions? It’s very simple: for example, we can replace an irrational expression with an expression or vice versa. That is, if the expression being converted contains an expression that matches in appearance the expression from the left (right) part of any of the listed properties of 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 roots.
Let's give 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 in different ways. For example, a root based on a property can be represented as , and a root using a property with k=3 - as , with this approach the solution will look like this: 
One could do it differently by replacing with , and then with , in which case the solution would look like this: 
Other solutions are possible, for example:
Let's look at the solution to another example. Let's transform the expression. Looking at the list of properties of roots, we select from it the properties we need to solve the example; it is clear that two of them are useful here and , which are valid for any a . We have: 
Alternatively, one could first transform the radical expressions using 
and then apply the properties of the roots
Up to this point, we have converted expressions that only contain square roots. It's time to work with roots that have different indicators.
Example.
Convert the irrational expression 
.
Solution.
By property 
the first factor of a given product can be replaced by the number −2: 
Go ahead. By virtue of the property, the second factor can be represented as , and it would not hurt to replace 81 with a quadruple power of three, since in the remaining factors the number 3 appears under the signs of the roots: 
It is advisable to replace the root of a fraction with a ratio of roots of the form , which can be transformed further: 
. We have
After performing operations with twos, the resulting expression will take the form , and all that remains is to transform the product of the roots.
To transform products of 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. Equality, which is applied from right to left, allows us to cope with this task. So . Taking this result into account, we have 
Now the product of roots can be replaced by the root of the product and perform the remaining, already obvious, transformations: 
Let's write a short version of the solution: 
Answer:
.
We emphasize separately that in order to apply the properties of 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 may cause incorrect results. For example, we know that the property holds for non-negative a . Based on it, we can easily move, 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 property indicated above, replace it with , then we actually replace −2 with 2. Indeed, ah. That is, for negative a the equality may be incorrect, just as other properties of roots may be incorrect 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 signs of the roots cannot be transformed using the properties of the roots. They just need to be “prepared” first by applying the rules for operating with numbers or using the definition of an odd root of 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 further apply the property of the root of a product: 
. And in one of the previous examples, it was necessary to move from root to root of the eighteenth power not like this, but like this 
.
So, to transform expressions using the properties of roots, you need
- 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 radical signs
To transform irrational expressions containing not only numbers but also variables under the root sign, the properties of roots listed in the first paragraph of this article must be applied carefully. This is mostly due to the conditions that the numbers involved in the formulas must satisfy. For example, based on the formula, the expression can be replaced by an expression only for those values of x that satisfy the conditions x≥0 and x+1≥0, since the specified formula is specified for a≥0 and b≥0.
What are the dangers 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 at x=−2. If we immediately substitute the number −2 instead of the variable x, we will get the value we need 
. 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 for x and arrive at the expression 
, which doesn't make sense.
Let's see what happens to the range of permissible values (APV) of the variable x when moving from expression to expression. It was not by chance that we mentioned the ODZ, since this is a serious tool for monitoring the admissibility of the transformations made, and a change in the ODZ after transforming an expression should, at a minimum, raise red flags. Finding the ODZ for these expressions is not difficult. For the expression ODZ is determined from the inequality x·(x+1)≥0, its solution gives the numerical set (−∞, −1]∪∪∪)
