Questions: Finding the nth root of a perfect nth power fraction Simplify. sqrt[3](8/27) Be sure to write your answer in lowest terms.

Finding the nth root of a perfect nth power fraction Simplify. sqrt[3](8/27) Be sure to write your answer in lowest terms.
Transcript text: Finding the $n^{\text {th }}$ root of a perfect $n^{\text {th }}$ power fraction Simplify. \[ \sqrt[3]{\frac{8}{27}} \] Be sure to write your answer in lowest terms.
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Solution

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Solution Steps

To simplify the expression \(\sqrt[3]{\frac{8}{27}}\), we need to find the cube root of both the numerator and the denominator separately. Since both 8 and 27 are perfect cubes, we can simplify the expression by taking the cube root of each.

Step 1: Finding the Cube Roots

We start with the expression \(\sqrt[3]{\frac{8}{27}}\). To simplify this, we find the cube roots of the numerator and the denominator separately: \[ \sqrt[3]{8} = 2 \quad \text{and} \quad \sqrt[3]{27} = 3 \]

Step 2: Forming the Simplified Fraction

Now that we have the cube roots, we can express the simplified fraction as: \[ \frac{\sqrt[3]{8}}{\sqrt[3]{27}} = \frac{2}{3} \]

Final Answer

Thus, the simplified form of \(\sqrt[3]{\frac{8}{27}}\) is \[ \boxed{\frac{2}{3}} \]

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