Question

Represent the following mixed infinite decimal periodic fractions as common fractions:
Simplify the following expressions.
$\frac{(\sqrt[5]{5a^{4/3}}{)}^{3/2}}{(\sqrt[5]{5a^4}{)}^3}\cdot \frac{(\sqrt{a\sqrt[3]{3a^{2}b}}{)}^4}{(\sqrt[3]{3a\sqrt{b}}{)}^4}$

Answer 1

The given equation is

$\frac{{\left(\sqrt[5]{5a^{4/3}}\right)}^{3/2}}{{\left(\sqrt[5]{5a^4}\right)}^3}\frac{{\left(\sqrt{a^3\sqrt{a^{2}b}}\right)}^4}{{\left(\sqrt[3]{3a\sqrt{b}}\right)}^4}$

$\frac{{\left(\sqrt[5]{5a^{4/3}}\right)}^{3/2}}{{\left(\sqrt[5]{5a^4}\right)}^3}\frac{{\left(\sqrt{a^3\sqrt{a^{2}b}}\right)}^4}{{\left(\sqrt[3]{3a\sqrt{b}}\right)}^4}\Rightarrow =\frac{a^{(\frac{4}{3\times 5}\times \frac{3}{2})}}{a^{\left(\frac{4\times 3}{5}\right)}}\frac{a^{(\frac{3\times 4}{2}+\frac{2\times 4}{2\times 2})}{b}^{\frac{4}{2\times 2}}}{a^{4/3}{b}^{\frac{4}{2\times 2}}}$

Now add all the powers of a and b we get

$\Rightarrow \frac{a^{(\frac{4}{3\times 5}\times \frac{3}{2})}}{a^{\left(\frac{4\times 3}{5}\right)}}\frac{a^{(\frac{3\times 4}{2}+\frac{2\times 4}{2\times 2})}{b}^{\frac{4}{2\times 2}}}{a^{4/3}{b}^{\frac{4}{2\times 2}}}=\frac{a^{2/5}}{a^{12/5}}\frac{a^{8}b}{a^{4/3}b}=a^{(\frac{2}{5}+8-\frac{12}{5}-\frac{4}{3})}{b}^0$

$=a^{14/3}$