Question

Which is equivalent to 80 Superscript one-fourth x?

Answer 1

Explanation:

I'm assuming you provided the equation:
`80^{(1)/(4)}x`

In general any nth root can be rewritten as:
`\sqrt[n]{x} = x^{(1)/(n)}`

We can refer to the law of exponents, to prove why this is the case.

You may recall from the law of exponents, the power of a power rule:
`(x^a)^b=x^(ab)`

So if we take the value:
`x^(1)/(n)` and raise it to the nth power as such:
`(x^(1)/(n))^n`, we can multiply the exponents to get:
`x^(n)/(n)=x^1=x`

This is by definition the nth root, if we can raise this to the nth power to get the original value.

So we can rewrite raising x to the one-fourth, as the fourth root of 80.

From here we get:
`x\sqrt[4]{80}`

Assuming your equation was actually:
`80^{(1)/(4)x}`

We can actually use our power to a power rule, since it goes both ways.

We can rewrite:
`80^{(1)/(4)x}\implies (80^x)^(1)/(4)`

From here we can use our nth root definition which we proved above to get the following:
`\sqrt[4]{80^x}`

Note: We could've rewritten the expression as such
`80^{(1)/(4)x}\implies (80^(1)/(4))^x`, since multiply the exponents would give us our original expression, with further simplification giving us:
`(\sqrt[4]{80})^x`, and this would give us the same expression, both are equally valid (except in certain cases where the domain may change as a result)