Question

Write \frac{\sqrt[4]{v^{9}}}{\sqrt[3]{v^{4}}}

3

v
4

​

4

v
9

​

​
as a single radical using the smallest possible root.

Answer 1

`\mbox {\large \boxed{\sqrt[12]{v^(11) }}}\\`

Explanation:

`\textrm{We are asked to simplify\:$\frac{\sqrt[4]{v^9}}{\sqrt[3]{v^4}}$}`

Using the fact that
`\sqrt[n]{x} = n^{{1/n}}`

`\mbox{\large {\sqrt[4]{v^9}} =\left(v^9)^(1/4) = v^(9/4)}\\\\`

`\mbox{\large {\sqrt[3]{v^4}}} = (v^4)^(1/3) = v^(4/3)}`

`\mbox{\large $\frac{\sqrt[4]{v^9}}{\sqrt[3]{v^4}}$}`
`= \mbox{\large (v^(9/4))/(v^(4/3))}`

Using the fact that
`(x^a)/(x^b) = x^(a-b)`

`\mbox{\large (v^(9/4))/(v^(4/3)) = v^(9/4 - 4/3)}`

`(9)/(4) - (4)/(3) = (9 \cdot 3 - 4 \cdot 4)/(12) = (27-16)/(12) = (11)/(12)`

`\mbox{\large (v^(9/4))/(v^(4/3)) = v^(9/4 - 4/3)= v^(11/12) } = \sqrt[12]{v^(11) }}`

Answer

`\mbox {\large \boxed{\sqrt[12]{v^(11) }}}\\`