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.
RootIndex 14 StartRoot x squared y Superscript 10 EndRoot
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Part 1
RootIndex 14 StartRoot x squared y Superscript 10 EndRootequals
  
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Answer 1

To simplify the expression \(\sqrt[14]{x^2 y^{10}}\) using rational exponents, we can rewrite the expression using the exponent notation.

The expression can be represented as:

\[
\sqrt[14]{x^2 y^{10}} = (x^2 y^{10})^{\frac{1}{14}}
\]

Applying the exponent to both terms inside the brackets, we have:

\[
= x^{2 \cdot \frac{1}{14}} y^{10 \cdot \frac{1}{14}} = x^{\frac{2}{14}} y^{\frac{10}{14}}
\]

Now we can simplify the exponents:

\[
= x^{\frac{1}{7}} y^{\frac{5}{7}}
\]

Now, we can convert back to radical notation. The expression in radical notation becomes:

\[
= \sqrt[7]{x} \cdot \sqrt[7]{y^5}
\]

Therefore, we can also write the expression in a simpler form as:

\[
= \frac{\sqrt[7]{y^5}}{\sqrt[7]{x}}
\]

So, the final simplified expression in radical notation is:

\[
\sqrt[14]{x^2 y^{10}} = \sqrt[7]{xy^5}
\]

For clarity, the answer can be expressed as:

\[
\sqrt[14]{x^2 y^{10}} = \sqrt[7]{x y^5}
\]