Question

if $7^a = sqrt8{7^3}$, what is the value of $a$ ?
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Explanation:

Step1: Rewrite the radical as an exponent

Recall that the \(n\)-th root of a number \(x\) can be written as \(x^{\frac{1}{n}}\). So, \(\sqrt[8]{7^{3}}\) can be rewritten as \(7^{\frac{3}{8}}\) because \(\sqrt[n]{x^{m}}=x^{\frac{m}{n}}\).

Step2: Equate the exponents

We are given that \(7^{a}=\sqrt[8]{7^{3}}\), and from Step 1 we know that \(\sqrt[8]{7^{3}} = 7^{\frac{3}{8}}\). Since the bases are the same (both are 7) and the exponential functions with the same base are equal when their exponents are equal, we can set the exponents equal to each other. So, \(a=\frac{3}{8}\).

Answer:

\(\frac{3}{8}\)