Question

Which of the following is the simplified form of fifth root of x times the fifth root of x times the fifth root of x times the fifth root of x?

Answer 1

Answer:

x^4/5

Step-by-step explanation:

Answer 2

Answer:

Simplified form of fifth root of x times the fifth root of x times the fifth root of x times the fifth root of x is $\sqrt[5]{x^{4}}$

Solution:

Need to find simplified form of fifth root of x times the fifth root of x times the fifth root of x times the fifth root of x. That is need to find simplified form of following expression.

$\sqrt[5]{x} \times \sqrt[5]{x} \times \sqrt[5]{x} \times \sqrt[5]{x}$

$\text { since } \sqrt[n]{a}=(a)^{\frac{1}{n}}$ we get

$=>(x)^{\frac{1}{5}} \times(x)^{\frac{1}{5}} \times(x)^{\frac{1}{5}} \times(x)^{\frac{1}{5}}$

Now using law of exponent that is $\mathrm{a}^{\mathrm{m}} \times \mathrm{a}^{\mathrm{n}}=\mathrm{a}^{\mathrm{m}+\mathrm{n}}$

$\begin{array}{l}{\Rightarrow(x)^{\frac{1}{5}} \times(x)^{\frac{1}{5}} \times(x)^{\frac{1}{5}} \times(x)^{\frac{1}{5}}} \\\\ {=(x)^{\frac{1}{5}+\frac{1}{5}} \times(x)^{\frac{1}{5}}} \\\\ {=(x)^{\frac{2}{5}} \times(x)^{\frac{2}{5}}} \\\\ {=(x)^{\frac{2}{5}} \times(x)^{\frac{2}{5}}} \\\\ {=(x)^{\frac{2}{5} +\frac{2}{5} \\\\ {=(x)^{\frac{4}{5}}}\end{array}$

Using another law of exponent that is $(a)^{m \times n}=\left((a)^{m}\right)^{n}$ we get

$\begin{array}{l}{(x)^{\frac{4}{5}}=(x)^{4 \times \frac{1}{5}}=\left((x)^{4}\right)^{\frac{1}{5}}} \\\\ {\left((x)^{4}\right)^{\frac{1}{5}}=\sqrt[5]{x^{4}}}\end{array}$

Hence the simplified form of fifth root of x times the fifth root of x times the fifth root of x times the fifth root of x is $\sqrt[5]{x^{4}}$