Question

What is the value of $[\sqrt[3]{39\sqrt{a^9}}{]}^4$ $[\sqrt[6]{63\sqrt{a^9}}{]}^4$

Answer 1

Given,

${\left[\sqrt[3]{39\sqrt{a^9}}\right]}^4{\left[\sqrt[6]{63\sqrt{a^9}}\right]}^4$

$={\left[\sqrt[3]{39a^{\frac{9}{2}}}\right]}^4{\left[\sqrt[6]{63a^{\frac{9}{2}}}\right]}^4$

$={\left[9^{\frac{1}{3}}{a}^{\frac{9}{2}\times \frac{1}{3}}\right]}^4{\left[3^{\frac{1}{6}}{a}^{\frac{9}{2}\times \frac{1}{6}}\right]}^4$

$={\left[9^{\frac{1}{3}}{a}^{\frac{3}{2}}\right]}^4{\left[3^{\frac{1}{6}}{a}^{\frac{3}{4}}\right]}^4$

$=\left[9^{\frac{4}{3}}{a}^{\frac{12}{2}}\right]\left[3^{\frac{4}{6}}{a}^{\frac{12}{4}}\right]$

$=\left[3^{\frac{8}{3}}{a}^6\right]\left[3^{\frac{2}{3}}{a}^3\right]$

$=3^{\frac{8}{3}+\frac{2}{3}}{a}^{6+3}$

$=3^{\frac{10}{3}}{a}^9$