Question

radicals and rational exponents: foun
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if $3^a = \sqrt5{3^4}$, what is the value of $a$?
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Explanation:

Step1: Recall radical to exponent rule

The \(n\)-th root of a number \(x\) can be written as \(x^{\frac{1}{n}}\). So, \(\sqrt[5]{3^{4}}\) can be rewritten using this rule.
\(\sqrt[5]{3^{4}} = 3^{\frac{4}{5}}\)

Step2: Equate exponents (since bases are equal)

We have the equation \(3^{a}=\sqrt[5]{3^{4}}\), and from Step 1, \(\sqrt[5]{3^{4}} = 3^{\frac{4}{5}}\). When two exponential expressions with the same base are equal, their exponents must be equal. So, if \(3^{a}=3^{\frac{4}{5}}\), then \(a = \frac{4}{5}\).

Answer:

\(\frac{4}{5}\)