Question

question 7
1 pts
evaluate.
512^{\frac{3}{4}}

Explanation:

Step1: Recall exponent rule

For \(a^{\frac{m}{n}}\), it can be rewritten as \((a^{\frac{1}{n}})^m\) or \(\sqrt[n]{a^m}\). Here, \(a = 512\), \(m = \frac{2}{3}\)? Wait, no, the exponent is \(\frac{2}{3}\)? Wait, the problem is \(512^{\frac{2}{3}}\)? Wait, the user wrote \(512^{\frac{2}{3}}\)? Wait, let's check: \(512\) is \(2^9\), so let's use the exponent rule \(a^{\frac{m}{n}}=\sqrt[n]{a^m}=(\sqrt[n]{a})^m\).

First, find the cube root of 512. Since \(8^3 = 512\) (because \(8\times8\times8 = 512\)), so \(\sqrt[3]{512}=8\).

Step2: Raise to the power of 2

Now, we have \((\sqrt[3]{512})^2 = 8^2\). Calculating \(8^2\), we get \(64\).

Alternatively, using \(a^{\frac{m}{n}}=a^{m\times\frac{1}{n}}\), so \(512^{\frac{2}{3}}=(2^9)^{\frac{2}{3}}\). Using the power of a power rule \((a^m)^n=a^{m\times n}\), we have \(2^{9\times\frac{2}{3}} = 2^6\) (since \(9\times\frac{2}{3}=6\)). And \(2^6 = 64\).

Answer:

\(64\)