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.
left parenthesis RootIndex 5 StartRoot xy EndRoot right parenthesis Superscript 10
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Part 1
left parenthesis RootIndex 5 StartRoot xy EndRoot right parenthesis Superscript 10equals
  
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Answer 1

To simplify the expression \(\left(\sqrt[5]{\sqrt{xy}}\right)^{10}\), we can start by using rational exponents.

1. **Convert the expression to rational exponents**:
\[
\sqrt[5]{\sqrt{xy}} = (xy)^{1/2} = (xy)^{1/2 \cdot 1/5} = (xy)^{1/10}
\]
So, the expression simplifies to:
\[
\left((xy)^{1/10}\right)^{10}
\]

2. **Use the power of a power property**: Multiply the exponents.
\[
(xy)^{1/10 \cdot 10} = (xy)^{1} = xy
\]

So, the simplified expression is:
\[
\left(\sqrt[5]{\sqrt{xy}}\right)^{10} = xy
\]

Now, writing it back in radical notation, since we simplified fully and the result is a product of variables, there's no further radical notation necessary.

The final answer is:
\[
\boxed{xy}
\]