Sure, let's simplify the given expression [tex]\(\sqrt{64y^5}\)[/tex] step by step.
1. Break the expression under the square root into separate parts:
[tex]\[
\sqrt{64y^5} = \sqrt{64} \cdot \sqrt{y^5}
\][/tex]
2. Simplify the square root of the numerical coefficient:
[tex]\[
\sqrt{64} = 8
\][/tex]
since [tex]\(64\)[/tex] is a perfect square ([tex]\(8^2 = 64\)[/tex]).
3. Simplify the square root of the variable part:
[tex]\[
\sqrt{y^5} = \sqrt{y^4 \cdot y} = \sqrt{(y^2)^2 \cdot y}
\][/tex]
Since taking the square root of [tex]\(y^4\)[/tex] is straightforward because it is a perfect square (i.e., [tex]\((y^2)^2\)[/tex]), we have:
[tex]\[
\sqrt{y^4 \cdot y} = \sqrt{(y^2)^2 \cdot y} = y^2 \cdot \sqrt{y}
\][/tex]
4. Combine the simplified parts:
[tex]\[
\sqrt{64y^5} = 8 \cdot y^2 \cdot \sqrt{y}
\][/tex]
Therefore, the simplified form of [tex]\(\sqrt{64y^5}\)[/tex] is:
[tex]\[
8 y^2 \sqrt{y}
\][/tex]
So the step-by-step simplified form is [tex]\(8 y^2 \sqrt{y}\)[/tex], and among the provided options, none directly match this simplified form. However, we followed the correct algebraic approach to simplify the expression.
