Question

Use rational exponents to simplify the expression. If rational exponents appear after​ simplifying, write the answer in radical notation. Assume that all variables represent positive numbers.
StartRoot 7 EndRoot times RootIndex 6 StartRoot 7 EndRoot
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Part 1
StartRoot 7 EndRoot times RootIndex 6 StartRoot 7 EndRootequals
  
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Answer 1

To simplify the expression \( \sqrt{7} \times \sqrt[6]{7} \) using rational exponents, we start by expressing each root in terms of exponents.

1. The square root \( \sqrt{7} \) can be written as \( 7^{1/2} \).
2. The sixth root \( \sqrt[6]{7} \) can be written as \( 7^{1/6} \).

Now we can rewrite the expression:

\[
\sqrt{7} \times \sqrt[6]{7} = 7^{1/2} \times 7^{1/6}
\]

Next, we apply the property of exponents that states \( a^m \times a^n = a^{m+n} \):

\[
7^{1/2} \times 7^{1/6} = 7^{1/2 + 1/6}
\]

To add \( \frac{1}{2} \) and \( \frac{1}{6} \), we need a common denominator. The least common multiple of 2 and 6 is 6. We convert \( \frac{1}{2} \):

\[
\frac{1}{2} = \frac{3}{6}
\]

Now we can add:

\[
\frac{3}{6} + \frac{1}{6} = \frac{4}{6}
\]

Simplifying \( \frac{4}{6} \):

\[
\frac{4}{6} = \frac{2}{3}
\]

Thus, we have:

\[
7^{1/2 + 1/6} = 7^{2/3}
\]

Now, we can convert this back to radical notation. The exponent \( \frac{2}{3} \) means the cube root of \( 7^2 \):

\[
7^{2/3} = \sqrt[3]{7^2}
\]

Thus, the final simplified expression in radical notation is:

\[
\sqrt[3]{49}
\]

So, the answer is:

\[
\sqrt[3]{49}
\]