<s>[INST] Simplify $\left(ab \sqrt[4]{a^{3}/\sqrt{b\sqrt{b}}}\right)^{2}$ Question: 
Simplify 
$$ \left(ab \sqrt[4]{a^{3}/\sqrt{b\sqrt{b}}}\right)^{2}$$
Attempted solution:
Rewriting it to look a bit better:
$$\left(ab \sqrt[4]{\frac{a^{3}}{\sqrt{b\sqrt{b}}}}\right)^{2}$$
Rewriting the denominator by replacing square roots with powers of fractions:
$$\left(ab \sqrt[4]{\frac{a^{3}}{(b (b^{\frac{1}{2}}))^{\frac{1}{2}}}}\right)^{2}$$
Combining b:s in the denominator:
$$\left(ab \sqrt[4]{\frac{a^{3}}{b^{\frac{3}{4}}}}\right)^{2}$$
Distributing the 4th root:
$$\left(\frac{ab \cdot a^{\frac{12}{16}}}{b^{\frac{3}{16}}}\right)^{2}$$
Combining a:s with a:s and b:s with b:s:
$$\left(b^{\frac{13}{16}} a^{\frac{28}{16}}\right)^{2}$$
Squaring gives the final result:
$$b^{26} a^{56}$$
The answer turns out to be:
$$a^{\frac{7}{2}} b^{\frac{13}{8}}$$
This is quite far away from the result I reached. Where did I go wrong, and are there are key insights that are useful for solving questions with lots and lots of square, cube and higher roots?
 [/INST] we have $a^2b^2\sqrt{\frac{a^3}{b^{3/4}}}=\sqrt{\frac{a^4b^4a^3}{b^{3/4}}}=a^{7/2}(b^{13/4})^{1/2}=a^{7/2}b^{13/8}$
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