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Part 1
Use rational exponents to simplify the following​ radical, then convert back to radical notation. Assume that all variables represent positive numbers.
RootIndex 15 StartRoot x Superscript 5 EndRoot15x5

Answer 1

To simplify the expression \( \sqrt[15]{x^5} \) using rational exponents, we can rewrite it as \( (x^5)^{\frac{1}{15}} \).

Using the property of exponents that \( (a^m)^n = a^{m \cdot n} \), we have:

\[
(x^5)^{\frac{1}{15}} = x^{5 \cdot \frac{1}{15}} = x^{\frac{5}{15}} = x^{\frac{1}{3}}
\]

Now, we can convert \( x^{\frac{1}{3}} \) back to radical notation:

\[
x^{\frac{1}{3}} = \sqrt[3]{x}
\]

Thus, the simplified form in radical notation is:

\[
\sqrt[15]{x^5} = \sqrt[3]{x}
\]