C. Perfect Cubes and Cube Roots
Perfect cubes and cube roots relate to each other in much the same way as do perfect squares and square roots. The difference is that a perfect cube is a number that is formed by multiplying a factor by itself three times instead of twice as is the case with a perfect square.
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Perfect Cube
generated when three identical factors are multiplied together Cube Roots a factor that is multiplied by itself three times to generate a perfect cube |
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Example 3 |
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Show the relationship between the number 64 and its cube root.
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The following table shows a list of some perfect cubes and their cube roots:
| Cube Root | Multiplication | Perfect Cube |
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1
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1 × 1 × 1
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1
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−1
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(−1) × (−1) × (−1)
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−1
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«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mroot»«mrow»«mo»-«/mo»«mn»1«/mn»«/mrow»«mn»3«/mn»«/mroot»«mo»=«/mo»«mo»-«/mo»«mn»1«/mn»«/math»
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2
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2 × 2 × 2
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8
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−2
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(−2) × (−2) × (−2)
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−8
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«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mroot»«mrow»«mo»-«/mo»«mn»8«/mn»«/mrow»«mn»3«/mn»«/mroot»«mo»=«/mo»«mo»-«/mo»«mn»2«/mn»«/math»
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3
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3 × 3 × 3
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27
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4
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4 × 4 × 4
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64
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5
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5 × 5 × 5
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125
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You might have noticed that 1 is both a perfect square and a perfect cube. 
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Another way to think of perfect cubes and cube roots is by considering the length and volume of a cube.
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Example 4 |
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The following cube has a length of 6 units. Determine the volume of the cube.
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If a number is a perfect square or cube, this can be shown using prime factorization. This is particularly useful for larger numbers.
The following two videos show examples of how to use prime factorization to determine if a number is a perfect square or a perfect cube.
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