To determine which expressions are equivalent to [tex]\(\sqrt{\frac{36 a^8}{225 a^2}}\)[/tex], let's simplify this expression step-by-step:
1. Simplify the fraction inside the square root:
The original expression is [tex]\(\sqrt{\frac{36 a^8}{225 a^2}}\)[/tex].
2. Factor the numerical coefficients:
Notice that both 36 and 225 are perfect squares. Specifically, [tex]\(36 = 6^2\)[/tex] and [tex]\(225 = 15^2\)[/tex]. Thus, we can rewrite the fraction as:
[tex]\[
\frac{36 a^8}{225 a^2} = \frac{6^2 a^8}{15^2 a^2}
\][/tex]
3. Separate the numerical part from the variable part inside the fraction:
Rewrite the fraction as:
[tex]\[
\frac{6^2}{15^2} \cdot \frac{a^8}{a^2}
\][/tex]
4. Simplify the coefficients:
Since [tex]\(\frac{6^2}{15^2} = \left(\frac{6}{15}\right)^2 = \left(\frac{2}{5}\right)^2\)[/tex], we have:
[tex]\[
\left( \frac{2}{5} \right)^2
\][/tex]
5. Simplify the variables using the properties of exponents:
For the variables, [tex]\(\frac{a^8}{a^2} = a^{8-2} = a^6\)[/tex].
6. Combine the simplified parts:
Now we can recombine the parts as:
[tex]\[
\frac{6^2 a^8}{225 a^2} = \left( \frac{2 a^6}{5} \right)^2
\][/tex]
7. Apply the square root:
Finally, we take the square root of [tex]\(\left( \frac{2 a^6}{5} \right)^2\)[/tex] which results in:
[tex]\[
\sqrt{\left( \frac{2 a^6}{5} \right)^2} = \frac{2 a^6}{5}
\][/tex]
Thus, [tex]\(\sqrt{\frac{36 a^8}{225 a^2}} = \frac{2 a^6}{5}\)[/tex]. Therefore the equivalent expression is:
[tex]\[
2 \cdot \frac{\sqrt{a^6}}{5}
\][/tex]
Which can be written as [tex]\(2 \cdot \frac{a^3}{5}\)[/tex], since [tex]\(\sqrt{a^6} = a^3\)[/tex].
Given this simplified expression, the expressions listed might include various forms equivalent to [tex]\(\frac{2 a^6}{5}\)[/tex] or [tex]\(2 \cdot \frac{a^3}{5}\)[/tex].
Identify which of the answer choices A, B, C, D, E match this simplified expression and select those.
