Question

Solve it properly plzz

Answer 1

answer is provided in attached image

Answer 2

Have a look at the equality given below.

$\Large \displaystyle \text{ \sqrt[n]{a}=a^{\frac{1}{n}} }$

According to this,

$\displaystyle \large \text{ \sqrt[3]{a}=a^{\frac{1}{3}} } \\ \\ \\ \text{ \sqrt[6]{a}=a^{\frac{1}{6}} }$

Thus,

$\displaystyle \left[\sqrt[3]{\sqrt[6]{5^9}}\right]^4\left[\sqrt[3]{\sqrt[6]{5^9}}\right]^4 \\ \\ \\ \left[\sqrt[3]{(5^9)^{\frac{1}{6}}}\right]^4\left[\sqrt[3]{(5^9)^{\frac{1}{6}}}\right]^4 \\ \\ \\ \left[\sqrt[3]{5^{\frac{9}{6}}}\right]^4\left[\sqrt[3]{5^{\frac{9}{6}}}\right]^4 \\ \\ \\ \left[\sqrt[3]{5^{\frac{3}{2}}}\right]^4\left[\sqrt[3]{5^{\frac{3}{2}}}\right]^4$

$\left[\left(5^{\frac{3}{2}}\right)^{\frac{1}{3}}\right]^4\left[\left(5^{\frac{3}{2}}\right)^{\frac{1}{3}}\right]^4 \\ \\ \\ \left[5^{\frac{3}{2} \times \frac{1}{3}}\right]^4\left[5^{\frac{3}{2} \times \frac{1}{3}}\right]^4 \\ \\ \\ \left[5^{\frac{3}{2} \times \frac{1}{3}}\right]^4\left[5^{\frac{3}{2} \times \frac{1}{3}}\right]^4 \\ \\ \\ \left[5^{\frac{1}{2}}\right]^4\left[5^{\frac{1}{2}}\right]^4 \\ \\ \\ 5^{\frac{1}{2} \times 4}5^{\frac{1}{2} \times 4} \\ \\ \\ 5^2 \cdot 5^2 \\ \\ \\ \large \bold{5^4}$

Thus the answer is option (2).