Simplifying Cube Roots That Contain Integers
To simplify a cube-root radical, we look for perfect cube factors of the
radicand.
Example 1
Simplify:
![](./articles_imgs/1015/simpli61.gif)
Solution
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![](./articles_imgs/1015/simpli62.gif) |
Write the prime factorization of 250.
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![](./articles_imgs/1015/simpli63.gif) |
Group triples of like factors to form perfect cubes. |
![](./articles_imgs/1015/simpli64.gif) |
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![](./articles_imgs/1015/simpli65.gif) |
Write as a product of two radicals.
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![](./articles_imgs/1015/simpli66.gif) |
Simplify
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![](./articles_imgs/1015/simpli68.gif) |
Thus, in simplified form,
![](./articles_imgs/1015/simpli69.gif)
Note:
If you realize that 125 is the largest perfect
cube factor of 250, then you can write:
![](./articles_imgs/1015/simpli70.gif)
Example 2
Simplify:
![](./articles_imgs/1015/simpli71.gif)
Solution
To simplify this expression, we’ll use the
Division Property of Cube Roots to rewrite
the radical as a quotient of radicals. Then
we’ll simplify each of those radicals. |
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![](./articles_imgs/1015/simpli72.gif) |
Write the numerator and denominator
under separate radical symbols. |
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![](./articles_imgs/1015/simpli73.gif) |
Write -40 using a perfect cube factor, -8. |
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![](./articles_imgs/1015/simpli74.gif) |
Write the numerator as the product of two
radicals.
|
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![](./articles_imgs/1015/simpli75.gif) |
Simplify the cube roots of any perfect cubes.
|
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![](./articles_imgs/1015/simpli76.gif) |
Thus, in simplified form,
![](./articles_imgs/1015/simpli77.gif)
Note:
If you have difficulty seeing the largest
perfect cube that is a factor of -40 or 27,
write their prime factorizations.
![](./articles_imgs/1015/simpli78.gif)
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