To determine which of the given expressions are equivalent to [tex]\(\sqrt{\frac{36 a^8}{225 a^2}}\)[/tex], we can simplify the given expression and then compare it to each of the provided options.
1. Simplify the given expression:
[tex]\[
\sqrt{\frac{36 a^8}{225 a^2}}
\][/tex]
First, simplify the fraction under the square root:
[tex]\[
\frac{36 a^8}{225 a^2} = \frac{36}{225} \cdot \frac{a^8}{a^2} = \frac{36}{225} \cdot a^{8-2} = \frac{36}{225} \cdot a^6
\][/tex]
Next, simplify the fraction:
[tex]\[
\frac{36}{225} = \frac{36 \div 9}{225 \div 9} = \frac{4}{25}
\][/tex]
So, the expression simplifies to:
[tex]\[
\sqrt{\frac{4}{25} a^6} = \sqrt{\frac{4}{25}} \cdot \sqrt{a^6} = \frac{2}{5} \cdot a^3 = \frac{2}{5} a^3
\][/tex]
Therefore, [tex]\(\sqrt{\frac{36 a^8}{225 a^2}} = \frac{2}{5} a^3\)[/tex].
Now, compare this with each of the given expressions:
2. Compare with each option:
- Option A:
[tex]\[
\sqrt{\frac{2(2)(3)(3)(\alpha)(\alpha)(\alpha)(\alpha)(\alpha)(\alpha)(\alpha)(\alpha)}{3(3)(5)(5)(\alpha)(\alpha)}} = \sqrt{\frac{4 \cdot 9 \cdot \alpha^8}{9 \cdot 25 \cdot \alpha^2}} = \sqrt{\frac{36 \alpha^8}{225 \alpha^2}}
\][/tex]
This simplifies to the original expression, so:
[tex]\[
\sqrt{\frac{36 a^8}{225 a^2}} = \frac{2}{5} a^3
\][/tex]
Thus, option A is equivalent.
- Option B:
[tex]\[
\sqrt{\frac{40^6}{25}}
\][/tex]
This does not simplify directly into [tex]\(\frac{2}{5} a^3\)[/tex]. So, option B is not equivalent.
- Option C:
[tex]\[
\frac{6}{25} \sqrt{\frac{a^9}{a^2}} = \frac{6}{25} \sqrt{a^{9-2}} = \frac{6}{25} \sqrt{a^7}
\][/tex]
This does not simplify directly to [tex]\(\frac{2}{5} a^3\)[/tex]. So, option C is not equivalent.
- Option D:
[tex]\[
\frac{6}{15} a^4 = \frac{2}{5} a^4
\][/tex]
This is not equivalent to [tex]\(\frac{2}{5} a^3\)[/tex]. So, option D is not equivalent.
- Option E:
[tex]\[
\frac{2}{5} a^3
\][/tex]
This is exactly the simplified form of the given expression. So, option E is equivalent.
3. Conclusion:
The expressions that are equivalent to [tex]\(\sqrt{\frac{36 a^8}{225 a^2}}\)[/tex] are:
- Option A: [tex]\(\frac{2}{5} a^3\)[/tex]
- Option E: [tex]\(\frac{2}{5} a^3\)[/tex]
Thus, the correct answers are A and E.
