Question

write the radical using a rational exponent.\\(\sqrt{2^{-7}}\\)\\(\sqrt{2^{-7}} = \square\\)(simplify your answer. use integers or fractions for any numbers in the expre

Explanation:

Step1: Recall the radical to exponent rule

The square root (which is a radical with index 2) of a number \(a\) can be written as \(a^{\frac{1}{2}}\). So for \(\sqrt{2^{-7}}\), we can apply the rule \(\sqrt[n]{a^m}=a^{\frac{m}{n}}\), where \(n = 2\) (since it's a square root) and \(m=-7\).

Step2: Apply the rule

Using the formula \(\sqrt[n]{a^m}=a^{\frac{m}{n}}\), substitute \(n = 2\) and \(m=-7\) into it. We get \(2^{\frac{-7}{2}}\), which can also be written as \(2^{-\frac{7}{2}}\) (since \(\frac{-7}{2}=-\frac{7}{2}\)).

Answer:

\(2^{-\frac{7}{2}}\) (or \(2^{\frac{-7}{2}}\))