Let’s compare cube roots.
Remember that square roots of whole numbers are defined as side lengths of squares. For example, √17 is the side length of a square whose area is 17. We define cube roots similarly, but using cubes instead of squares. The number ∛17, pronounced “the cube root of 17,” is the edge length of a cube which has a volume of 17.
We can approximate the values of cube roots by observing the whole numbers around it and remembering the relationship between cube roots and cubes. For example, ∛20 is between 2 and 3 since 23 = 8 and 33 = 27, and 20 is between 8 and 27. Similarly, since 100 is between 43 and 53, we know ∛100 is between 4 and 5. Many calculators have a cube root function which can be used to approximate the value of a cube root more precisely. Using our numbers from before, a calculator will show that ∛20 ≈ 2.7144 and that ∛20 ≈ 4.6416.
Also like square roots, most cube roots of whole numbers are irrational. The only time the cube root of a number is a whole number is when the original number is a perfect cube.
Decide if each statement is true or false.
What two whole numbers does each cube root lie between? Be prepared to explain your reasoning.
The numbers x, y, and z are positive, and:
Diego knows that 82 = 64 and that 43 = 64. He says that this means the following are all true:
√64 = 8 is true because 82 = 64
∛64 = 4 is true because 43 = 64
√-64 = -8 is not true because (-8)2 = 64 and not -64
∛-64 = -4 is true because (-4)3 = -64
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