Simplifying radicals: Section 2
We can identify 
with the absolute value of x (Lesson 12).
.
For, when x ≥ 0, then
.
But if x < 0 -- if, for example, x = −5 -- then
because the square root is never negative. (Lesson 26.) Rather, when x < 0, then
.
.
Therefore in general we must write
.
conforms to the definition of the absolute value.
Example 4. Compare (
)² and .
()² = x. (Lesson 26.) For, in order for that radical to be a real number, the radicand x may not be negative.
= x -- only if x ≥ 0. For any value of x, we must write
.
Example 5. Since the square of any power produces an even exponent --
(a3)2 = a6
-- then the square root of an even power will be half the exponent.
= a3.
As for an odd power, such as a7, it is composed of an even power times a:
a7 = a6a.
Therefore,
= 
= a3
.
(These results hold only for a ≥ 0.)
Problem 5. Simplify each radical. (Assume a ≥ 0.)
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| a) |  = a² |
b) |  = a5 |
c) |  = an |
| d) | 
| = |  |
= | a |
e) |  =  =
a4
| |||||
| f) | 
| = |  |
= | a7 |
g) | 
| = |  |
= | an |
|
Note: '2n' in algebra, as in part c), indicates an even number, that is, a multiple of 2. The variable n typically signifies an integer. We signify an odd number, then, as '2n + 1,' as in part g).
Problem 6. Simplify each radical. Remove the even powers. (Assume that the variables do not have negative values.)
| a) |  =  |
= | 2x |
| b) |  =  |
= | 2x²y3 |
| c) |  =  |
= | 3x4yz² |
Factors of the radicand
Problem 7. True or false? That is, which of these is a rule of algebra? (Assume that a and b do not have negative values.)
a)  |
True. This is the rule, and the only one. The square root of a product is the product of the square roots of each factor. |
b)  = +  |
False. The radicand is not made up of factors, as in part a). |
c)  = a + b. |
False! The radicand is not made up of factors. |
d)  = a. |
True. |
e)  = a + b. |
True. The radicand is (a + b)². |
Problem 8. Express each radical in simplest form.
a) 
=
= 2
.
To simplify a radical, the radicand must be composed of factors!
b) 
=
= 2a
c) 
=
= 3b
Fractional radicand
A radical is in its simplest form when the radicand is not a fraction.
Example 6. The denominator a square number. When the denominator is a square number, as 
, then
|
= | 1 2 |
In general,
For, a· a = a².
Example 7.
 |
= |  |
The definition of division | |
| = | 1 2 |
 |
||
Problem 9. Simplify each radical.
a)  = |
1 3 |
b)  = |
 |
= | 1 3 |
 |
c)  = |
 |
= | 2 5 |
 |
d)  = |
 |
= | 5 6 |
 |
| Example 8. | The denominator not a square number. |
Simplify  .
| |
Solution. When the denominator is not a square number, we can make it a square number by multiplying it. In this example, we will multiply it by itself, that is, by 2. But then we must multiply the numerator also by 2:
|
= |  |
= |  |
= | 1 2 |
 . |
Example 9. Simplify 
.
Solution. The denominator must be a perfect square. We can make 50 into a square number simply by multiplying by 2. We can make x a square by multiplying by x. And y² is already a square. Therefore,
Example 10. Simplify 
. (Assume that the variables do not have negative values.)
Solution. Again, the denominator must be a perfect square. It must be composed of even powers. Therefore, make the denominator into a product of even powers simply simply by multiplying it -- and the numerator -- by bc. Then extract half of the even powers.
Problem 10. Simplify each radical. (Assume that the variables do not have negative values.)
a)  = |
 |
= |
1 3 |
|
b)  = |
 |
= |
1 5 |
|
c)  = |
 |
= |
1 7 |
 |
d)  = |
 |
= |  |
= |
5 6x |
 |
e)  = |
 |
= |
2 x² |
 |
f)  = |
 |
= |
a² bc² |
 |
| Problem 11. Show |  |
A problem that asks you to show, means to write what's on the left, and then transform it algebraically so that it looks like what's on the right.
| Solution. |  |
= |  |
= |  |
= |  |
Next Lesson: Multipying and dividing radicals
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