When are we suppose to Rationalize the denominator? Like here, 1/\sqrt[4]{x^3} Is that the final answer? How come we can't rationalize this denominator? And other example--…

Question

When are we suppose to Rationalize the denominator?

Like here, 1/
`\sqrt[4]{x^3}` Is that the final answer? How come we can't rationalize this denominator?
And other example---- x^-1/5 and my teacher said the answer for this one is - 1/
`\sqrt[5]{x}` How come? Can't we rationalize the denominator?
This is so hard? Can somebody help please?

Answer 1

it is encouraged to rationalize the denomenator
you can rationalize the denomenator

one way is to convert to exponent
and remember your exponential rules
remember that
`\sqrt[n]{x^m}=x^{(m)/(n)}`
also,
`(x^a)(x^b)=x^(a+b)`
and
`x^(-m)=(1)/(x^m)`

so


`\frac{1}{\sqrt[4]{x^3}}=\frac{1}{x^{(3)/(4)}}`
so we want x^{4/4}, so 1/4+3/4=4/4
times the whole thing by
`\frac{x^{(1)/(4)}}{x^{(1)/(4)}}` to get

`\frac{x^{(1)/(4)}}{x^{(4)/(4)}}=\frac{x^{(1)/(4)}}{x}=\frac{\sqrt[4]{x}}{x}`
but, it looks alot nicer in the original form tho

the 2nd one, we multiply it by
`\frac{x^{(4)/(5)}}{x^{(4)/(5)}}` to get
`\frac{x^{(4)/(5)}}{x}=\frac{\sqrt[5]{x^4}}{x}`
but it looks nicer in original form tho



so you can ratinalize the denomenator but you don't always have to