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Which is equivalent to 80 Superscript one-fourth x?

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Answer:

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)

User Mikah
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