Question

question 1-5
evaluate the expression.
$4096^{\frac{1}{6}} = \square$

Explanation:

Step 1: Recall the definition of a fractional exponent

A fractional exponent \(a^{\frac{1}{n}}\) means the \(n\)-th root of \(a\), i.e., \(a^{\frac{1}{n}}=\sqrt[n]{a}\). So, \(4096^{\frac{1}{6}}=\sqrt[6]{4096}\).

Step 2: Find the number whose 6th power is 4096

We know that \(2^{12} = 4096\) (since \(2^{10}=1024\), \(2^{11}=2048\), \(2^{12}=4096\)). Also, \(2^{12}=(2^{2})^{6}\) because \((a^{m})^{n}=a^{mn}\), so \((2^{2})^{6}=4^{6}\). Therefore, \(\sqrt[6]{4096}=\sqrt[6]{4^{6}} = 4\).

Answer:

\(4\)