Question

Rewrite in simplest radical form x to the five sixths power divided by x to the one sixth power. Show each step of your process.

Answer 1

To simplify the expression \( \frac{x^{\frac{5}{6}}}{x^{\frac{1}{6}}} \), we can apply the properties of exponents. Here's how to do it step by step:

1. **Use the Quotient Rule of Exponents**: The quotient rule states that when you divide two expressions with the same base, you can subtract the exponents:
\[
\frac{a^m}{a^n} = a^{m-n}
\]

2. **Apply the Quotient Rule**: For our expression, we have:
\[
\frac{x^{\frac{5}{6}}}{x^{\frac{1}{6}}} = x^{\frac{5}{6} - \frac{1}{6}}
\]

3. **Subtract the Exponents**: Now, we subtract the exponents:
\[
\frac{5}{6} - \frac{1}{6} = \frac{5 - 1}{6} = \frac{4}{6}
\]

4. **Simplify the Fraction**: Now we simplify \( \frac{4}{6} \):
\[
\frac{4}{6} = \frac{2}{3}
\]

5. **Rewrite the Expression**: So we have:
\[
x^{\frac{5}{6} - \frac{1}{6}} = x^{\frac{2}{3}}
\]

6. **Convert to Radical Form**: The expression \( x^{\frac{2}{3}} \) can be expressed in radical form as:
\[
x^{\frac{2}{3}} = \sqrt[3]{x^2}
\]

Thus, the final result in simplest radical form is:
\[
\sqrt[3]{x^2}
\]