Let's simplify the given expression [tex]\(\sqrt[4]{36 a^2 b^8}\)[/tex] step-by-step, assuming all variables are positive real numbers.
1. Rewrite the expression using fractional exponents:
[tex]\[
\sqrt[4]{36 a^2 b^8} = (36 a^2 b^8)^{\frac{1}{4}}
\][/tex]
2. Distribute the fractional exponent to each part of the product inside the radical:
[tex]\[
(36 a^2 b^8)^{\frac{1}{4}} = 36^{\frac{1}{4}} \cdot (a^2)^{\frac{1}{4}} \cdot (b^8)^{\frac{1}{4}}
\][/tex]
3. Simplify each term individually:
- For [tex]\(36^{\frac{1}{4}}\)[/tex]:
[tex]\[
36 = 6^2 \implies 36^{\frac{1}{4}} = (6^2)^{\frac{1}{4}} = 6^{\frac{2}{4}} = 6^{\frac{1}{2}} = \sqrt{6}
\][/tex]
- For [tex]\((a^2)^{\frac{1}{4}}\)[/tex]:
[tex]\[
(a^2)^{\frac{1}{4}} = a^{2 \cdot \frac{1}{4}} = a^{\frac{1}{2}} = \sqrt{a}
\][/tex]
- For [tex]\((b^8)^{\frac{1}{4}}\)[/tex]:
[tex]\[
(b^8)^{\frac{1}{4}} = b^{8 \cdot \frac{1}{4}} = b^{2}
\][/tex]
4. Combine the simplified terms:
[tex]\[
\sqrt[4]{36 a^2 b^8} = \sqrt{6} \cdot \sqrt{a} \cdot b^2 = b^2 \sqrt{6a}
\][/tex]
So, the simplified expression is [tex]\(b^2 \sqrt{6 a}\)[/tex].
### Comparison with Multiple Choice Options:
- [tex]\(b^2 \sqrt{6 a}\)[/tex]
- [tex]\(36 b^8 \sqrt{a}\)[/tex]
- [tex]\(36 a^2 \sqrt{b}\)[/tex]
- [tex]\(b^2 \sqrt{a} \sqrt[4]{36}\)[/tex]
Based on our simplification process, the correct answer is:
[tex]\[ b^2 \sqrt{6 a} \][/tex]
