Simplifying Radicals Involving Variables
Examples
In the following example we start with a square root of a quotient.
Example 1
Radicals with variables
Simplify each expression. Assume all variables represent positive real numbers.
![](./articles_imgs/1030/simpli77.gif)
![](./articles_imgs/1030/simpli78.gif)
Solution
![](./articles_imgs/1030/simpli79.gif) |
![](./articles_imgs/1030/simpli80.gif) |
Quotient rule for radicals |
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![](./articles_imgs/1030/simpli81.gif) |
Rationalize the denominator. |
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![](./articles_imgs/1030/simpli82.gif) |
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![](./articles_imgs/1030/simpli83.gif) |
![](./articles_imgs/1030/simpli84.gif) |
Quotient rule for radicals |
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![](./articles_imgs/1030/simpli85.gif) |
Product rule for radicals |
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![](./articles_imgs/1030/simpli86.gif) |
Simplify. |
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![](./articles_imgs/1030/simpli87.gif) |
Rationalize the denominator. |
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![](./articles_imgs/1030/simpli88.gif) |
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Any variable with an exponent that is a multiple of 3 is a perfect cube.
For example, a3, b6, c15, and w39
are perfect cubes. Each of these expressions is the cube of a variable with an
integral exponent. For any values of the variables we can write
![](./articles_imgs/1030/simpli89.gif)
Note that when we find the cube root, the result has one-third of the
original exponent.
If the exponent on a variable is a multiple of 4, we have a perfect fourth
power; if the exponent is a multiple of 5, we have a perfect fifth power; and so
on. In the next example we simplify radicals with an index higher than 2.
Example 2
Simplifying higher-index radicals with variables
Simplify. Assume the variables represent positive numbers.
![](./articles_imgs/1030/simpli90.gif)
Solution
a) Use the product rule to place the largest perfect cube factors under the
first radical and the remaining factors under the second:
![](./articles_imgs/1030/simpli91.gif)
b) Place the largest perfect fourth power factors under the first radical and
the remaining factors under the second:
![](./articles_imgs/1030/simpli92.gif)
c) Multiply by
to
rationalize the denominator:
![](./articles_imgs/1030/simpli94.gif)
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