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Express ∛(1/2) × ∛(1/4) in its simplest form.

Express the cube root of one-half multiplied by the cube root of one-quarter in its simplest form.

In this question, we are asked to express the product of the cube root of two rational numbers in its simplest form. Since we want to simplify the product of the cube roots of two numbers, we can begin by recalling that for any real numbers 𝑎 and 𝑏, the cube root of 𝑎 times the cube root of 𝑏 is equal to the cube root of 𝑎 times 𝑏. Setting 𝑎 equal to one-half and 𝑏 equal to one-quarter gives us that the cube root of one-half multiplied by the cube root of one-quarter is equal to the cube root of one-half times one-quarter. We can then evaluate one-half times one-quarter by multiplying their numerators and denominators. This gives us one-eighth, so we need to find the cube root of one-eighth.

We could evaluate this directly by using the definition of a cube root. Or we could use the fact that for any real numbers 𝑥 and 𝑦 with 𝑦 not equal to zero, we have that the cube root of 𝑥 over 𝑦 equals the cube root of 𝑥 over the cube root of 𝑦. Applying this result with 𝑥 equals one and 𝑦 equals eight gives us the cube root of one over the cube root of eight. We can then calculate that the cube root of one is one and the cube root of eight is two, giving us that the simplest form of the cube root of one-half times the cube root of one-quarter is one-half.

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