To simplify the expression [tex]\(\sqrt[8]{a^4 b^2}\)[/tex], follow these steps:
1. Rewriting the Radicand:
The expression inside the 8th root is [tex]\(a^4 b^2\)[/tex]. Our goal is to simplify this expression under the 8th root.
2. Breaking Down the Exponent:
Recall that [tex]\(\sqrt[n]{x} = x^{\frac{1}{n}}\)[/tex]. Using this property, we can rewrite the 8th root of [tex]\(a^4 b^2\)[/tex] as [tex]\((a^4 b^2)^{\frac{1}{8}}\)[/tex].
3. Applying Exponent Rules:
Apply the exponent rule [tex]\((xy)^n = x^n y^n\)[/tex] to the expression [tex]\((a^4 b^2)^{\frac{1}{8}}\)[/tex]:
[tex]\[
(a^4 b^2)^{\frac{1}{8}} = a^{4 \cdot \frac{1}{8}} \cdot b^{2 \cdot \frac{1}{8}}
\][/tex]
4. Simplifying the Exponents:
Now, simplify the exponents:
[tex]\[
a^{4 \cdot \frac{1}{8}} = a^{\frac{4}{8}} = a^{\frac{1}{2}} = \sqrt{a}
\][/tex]
[tex]\[
b^{2 \cdot \frac{1}{8}} = b^{\frac{2}{8}} = b^{\frac{1}{4}} = \sqrt[4]{b}
\][/tex]
5. Combining the Results:
Combine the simplified expressions:
[tex]\[
(a^4 b^2)^{\frac{1}{8}} = \sqrt{a} \cdot b^{\frac{1}{4}}
\][/tex]
Thus, we have:
[tex]\[
\sqrt[8]{a^4 b^2} = \sqrt{a} \cdot b^{\frac{1}{4}}
\][/tex]
