Question

type the correct answer in each box. use numerals instead of words. the expression above can also be written in the form (sqrtb{a}). for this expression, (a =) (\boxed{}) and (b =) (\boxed{}).

Explanation:

Step1: Recall exponent-root identity

Recall that $x^m = \sqrt[n]{x^{mn}}$ or $x^{\frac{p}{q}} = \sqrt[q]{x^p}$. Here, $7^4$ is $7^{\frac{4}{1}}$, so we match to $\sqrt[b]{a}$.

Step2: Assign values to a and b

For $7^4 = \sqrt[b]{a}$, we can write $7^4 = \sqrt[1]{7^4}$ (since the 1st root of a number is itself).

Answer:

$a = 7^4 = 2401$ and $b = 1$

Alternatively, if interpreting as rewriting $7^4$ as a root with a different exponent:

Step1: Rewrite exponent as fraction

$7^4 = 7^{\frac{8}{2}} = \sqrt[2]{7^8}$, or $7^{\frac{12}{3}} = \sqrt[3]{7^{12}}$, but the simplest form is using the 1st root:

Step2: Identify base and root

The most direct match is $7^4 = \sqrt[1]{7^4}$, so: