Question Video: Simplifying Algebraic Expressions Involving Exponents and Cube Roots Mathematics

Simplify ∛(64𝑚³).

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Video Transcript

Simplify the cube root of 64 times 𝑚 cubed.

In order to be able to simplify an expression involving an 𝑛th root, where here 𝑛 is equal to three, we’ll recall one of the properties that applies to 𝑛th roots. This property tells us what happens when we multiply a pair of 𝑛th roots. Specifically, for positive real numbers 𝑎 and 𝑏, the 𝑛th root of 𝑎 times the 𝑛th root of 𝑏 is equivalent to the 𝑛th root of 𝑎𝑏. We’re going to apply this property in reverse. And it allows us to separate the cube root of 64𝑚 cubed into the product of the cube root of 64 and the cube root of 𝑚 cubed.

The next property we’re interested in tells us that if 𝑛 is an odd integer, which it is here, it’s three, then the 𝑛th root of 𝑎 all raised to the 𝑛th power is equal to the 𝑛th root of 𝑎 to the 𝑛th power, which is simply equal to 𝑎. And this is great. This allows us to simplify this part of the expression, the cube root of 𝑚 cubed. Since the root is odd, in other words, 𝑛 is equal to three, we can say that the cube root of 𝑚 cubed is simply equal to 𝑚. And of course, we know the value of the cube root of 64. It’s simply four. So we can substitute the cube root of 𝑚 cubed equals 𝑚 and the cube root of 64 equals four back into our earlier equation. And that will allow us to simplify the original expression.

When we do, we find that the cube root of 64 times the cube root of 𝑚 cubed is four times 𝑚, which can of course be fully simplified to four 𝑚. And so, we’ve simplified the cube root of 64𝑚 cubed. It’s four 𝑚.