Question Video: Simplifying the Multiplication and Division of Multiple Cube Roots Mathematics

Simplify (∛2 × ∛4)/(∛32 × ∛−2).

02:14

Video Transcript

Simplify the cube root of two multiplied by the cube root of four divided by the cube root of 32 multiplied by the cube root of negative two.

We begin by noting that none of the numbers here are perfect cubes. And as such, we cannot directly evaluate any of the individual radicals. Instead, we will use two properties of cube roots. Firstly, the cube root of 𝑎 multiplied by the cube root of 𝑏 is equal to the cube root of 𝑎 multiplied by 𝑏. This holds if 𝑎 and 𝑏 are real numbers, as in this case. We can rewrite the numerator of our fraction as the cube root of two multiplied by four. And the denominator simplifies to the cube root of 32 multiplied by negative two. This in turn gives us the cube root of eight over the cube root of negative 64.

Next, we use the fact that when 𝑎 and 𝑏 are real numbers and 𝑏 is nonzero, the cube root of 𝑎 over the cube root of 𝑏 is equal to the cube root of 𝑎 divided by 𝑏. We can therefore rewrite our expression as the cube root of eight over negative 64. The fraction eight over 64 written in its simplest form is one-eighth. This means that our expression simplifies to the cube root of negative one-eighth.

Finally, since negative one-half cubed is negative one-eighth, then the cube root of negative one-eighth is negative one-half. The expression the cube root of two multiplied by the cube root of four divided by the cube root of 32 multiplied by the cube root of negative two in its simplest form is negative one-half.

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