To find the simplest form of \(\sqrt{a^7}\), let's break down the exponent rules and manipulation of square roots.
First, recall that the square root of a power can be expressed as:
[tex]\[
\sqrt{a^b} = a^{b/2}
\][/tex]
Given the expression \(\sqrt{a^7}\), we can apply this rule:
[tex]\[
\sqrt{a^7} = a^{7/2}
\][/tex]
Next, express \(a^{7/2}\) in terms of an integer power times a square root. Notice that:
[tex]\[
a^{7/2} = a^{3.5} = a^{3+0.5} = a^3 \cdot a^{0.5}
\][/tex]
Since \(a^{0.5} = \sqrt{a}\), we can rewrite it as:
[tex]\[
a^{3+0.5} = a^3 \cdot \sqrt{a}
\][/tex]
So, the simplest form of \(\sqrt{a^7}\) is:
[tex]\[
a^3 \cdot \sqrt{a}
\][/tex]
Thus, the correct answer is:
[tex]\[
a^3 \sqrt{a}
\][/tex]
