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?

x to the 1 over fifth power
x to the 4 over fifth power
x to the four over twentieth power
x

Answer 1

`\large\boxed{x^(4)/(5)}`

Explanation:

`\sqrt[n]{a}=a^(1)/(n)\Rightarrow\sqrt[5]{x}=x^(1)/(5)\\\\\sqrt[5]{x}\cdot\sqrt[5]{x}\cdot\sqrt[5]{x}\cdot\sqrt[5]{x}=x^(1)/(5)\cdot x^(1)/(5)\cdot x^(1)/(5)\cdot x^(1)/(5)\qquad\text{use}\ a^n\cdot a^m=a^(n+m)\\\\=x^{(1)/(5)+(1)/(5)+(1)/(5)+(1)/(5)}=x^(4)/(5)`

Answer 2

`x^{(4)/(5)}`

Explanation:

fifth root of x can be written in exponential for as:

`x^(1)/(5)`

`x^(1)/(5)` times
`x^(1)/(5)` times
`x^(1)/(5)` times
`x^(1)/(5)`

WE apply exponential property to multiply it

a^m times a^n= a^{m+n}

`x^(1)/(5)` times
`x^(1)/(5)` times
`x^(1)/(5)` times
`x^(1)/(5)`

`x^{(1)/(5) +(1)/(5)+(1)/(5)+(1)/(5)}`

The denominator of the fractions are same so we add the numerators

`x^{(4)/(5)}`