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 18 StartRoot 2 y EndRoot right parenthesis Superscript 4
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Part 1
left parenthesis RootIndex 18 StartRoot 2 y EndRoot right parenthesis Superscript 4equals
  
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Answer 1

To simplify the expression \((\sqrt[18]{2y})^4\), we can use the properties of exponents and rational exponents.

1. **Express the root as a rational exponent:**
\[
\sqrt[18]{2y} = (2y)^{\frac{1}{18}}
\]

2. **Rewrite the initial expression using the rational exponent:**
\[
(\sqrt[18]{2y})^4 = \left((2y)^{\frac{1}{18}}\right)^4
\]

3. **Use the power of a power property of exponents (\((a^m)^n = a^{m \cdot n}\)) to multiply the exponents:**
\[
(2y)^{\frac{1}{18} \cdot 4} = (2y)^{\frac{4}{18}}
\]

4. **Simplify the exponent \(\frac{4}{18}\) to \(\frac{2}{9}\):**
\[
(2y)^{\frac{2}{9}} = (2y)^{\frac{2}{9}} = 2^{\frac{2}{9}}y^{\frac{2}{9}}
\]

5. **Now rewrite it back in radical notation:**
\[
2^{\frac{2}{9}} y^{\frac{2}{9}} = \sqrt[9]{2^2 y^2}
\]

Thus, the simplified expression is:

\[
\sqrt[9]{4y^2}
\]

So, the final answer is:
\[
\left(\sqrt[18]{2y}\right)^4 = \sqrt[9]{4y^2}
\]