Sure, let's break down the expression step by step:
We start with the expression:
[tex]\[
\left\{\sqrt{\left(a^2+\sqrt[3]{a^4 b^2}\right)}+\sqrt{\left(b^2+\sqrt[3]{a^2 b^4}\right)}\right\}^{\frac{2}{3}}
\][/tex]
1. Evaluate the terms inside the inner square roots:
- First term: [tex]\(a^2 + \sqrt[3]{a^4 b^2}\)[/tex]
- Second term: [tex]\(b^2 + \sqrt[3]{a^2 b^4}\)[/tex]
Notice that the cube root expressions [tex]\( \sqrt[3]{a^4 b^2} \)[/tex] and [tex]\( \sqrt[3]{a^2 b^4} \)[/tex].
2. Calculate the inner cube roots:
- [tex]\(\sqrt[3]{a^4 b^2} \)[/tex] can be rewritten as [tex]\((a^4 b^2)^{\frac{1}{3}}\)[/tex]
- [tex]\(\sqrt[3]{a^2 b^4} \)[/tex] can be rewritten as [tex]\((a^2 b^4)^{\frac{1}{3}}\)[/tex]
3. Simplify the cube roots:
- [tex]\((a^4 b^2)^{\frac{1}{3}} = a^{\frac{4}{3}} b^{\frac{2}{3}}\)[/tex]
- [tex]\((a^2 b^4)^{\frac{1}{3}} = a^{\frac{2}{3}} b^{\frac{4}{3}}\)[/tex]
4. Substitute these back into the expression:
- [tex]\(a^2 + a^{\frac{4}{3}} b^{\frac{2}{3}}\)[/tex]
- [tex]\(b^2 + a^{\frac{2}{3}} b^{\frac{4}{3}}\)[/tex]
Therefore, the terms inside the square roots now look like:
[tex]\[
\sqrt{a^2 + a^{\frac{4}{3}} b^{\frac{2}{3}}} \quad \text{and} \quad \sqrt{b^2 + a^{\frac{2}{3}} b^{\frac{4}{3}}}
\][/tex]
5. Sum the square roots:
[tex]\[
\sqrt{a^2 + a^{\frac{4}{3}} b^{\frac{2}{3}}} + \sqrt{b^2 + a^{\frac{2}{3}} b^{\frac{4}{3}}}
\][/tex]
6. Raise the result to the power of [tex]\(\frac{2}{3}\)[/tex]:
[tex]\[
\left( \sqrt{a^2 + a^{\frac{4}{3}} b^{\frac{2}{3}}} + \sqrt{b^2 + a^{\frac{2}{3}} b^{\frac{4}{3}}} \right)^{\frac{2}{3}}
\][/tex]
Thus, the final simplified form of the expression is:
[tex]\[
\boxed{\left( \sqrt{a^2 + a^{\frac{4}{3}} b^{\frac{2}{3}}} + \sqrt{b^2 + a^{\frac{2}{3}} b^{\frac{4}{3}}} \right)^{\frac{2}{3}}}
\][/tex]
This matches the given result with the numerical approximations for the exponents.
