Question

$\sqrt[5]{ {x}^{4} \sqrt[4]{ {x}^{3} \sqrt[3]{ {x}^{2} \sqrt{x} } } }$
​

Answer 1

$\sqrt[5]{ {x}^{4} \sqrt[4]{ {x}^{3} \sqrt[3]{ {x}^{2} \sqrt{x} } } }$

As we know

$\sqrt[q]{p} = p {}^{ \frac{1}{q} } \\ \\ {p}^{q} \times p {}^{r} = {p}^{q + r}$

Taking

$\sqrt[3]{ {x}^{2} \sqrt{x} } \\ \\ \sqrt[3]{ {x}^{2} \times {x}^{ \frac{1}{2} } } \\ \\ \sqrt[3]{ {x}^{2 + \frac{1}{2} } } \\ \\ \sqrt[ 3]{ {x}^{ \frac{5}{2} } } \\ \\ {x}^{ \frac{5}{2} \times \frac{1}{3} } \\ \\ {x}^{ \frac{5}{6} } \\ \\ \sqrt[3]{ {x}^{2} \sqrt{x} } = {x}^{ \frac{5}{6} }$

Now

$\sqrt[4]{ {x}^{3} \sqrt[3]{ {x}^{2} \sqrt{x} } } \\ \\ \sqrt[4]{ {x}^{3} \times {x}^{ \frac{5}{6} } } \\ \\ \sqrt[4]{ {x}^{3 + \frac{5}{6} } } \\ \\ \sqrt[4]{ {x}^{ \frac{23}{6} } } \\ \\ {x}^{ \frac{23}{6} \times \frac{1}{4} } \\ \\ {x}^{ \frac{23}{24} } \\ \\ \sqrt[4]{ {x}^{3} \sqrt[3]{ {x}^{2} \sqrt{x} } } = {x}^{ \frac{23}{24} }$

Now

$\sqrt[5]{ {x}^{4} \sqrt[4]{ {x}^{3} \sqrt[3]{ {x}^{2} \sqrt{x} } } } \\ \\ \sqrt[5]{ {x}^{4} \times {x}^{ \frac{23}{24} } } \\ \\ \sqrt[5]{ {x}^{4 + \frac{23}{24} } } \\ \\ \sqrt[5]{ {x}^{ \frac{119}{24} } } \\ \\ {x}^{ \frac{119}{24} \times \frac{1}{5} } \\ \\ {x}^{ \frac{119}{120} }$

$\sqrt[5]{ {x}^{4} \sqrt[4]{ {x}^{3} \sqrt[3]{ {x}^{2} \sqrt{x} } } }={x}^{\frac{119}{120}}$