Let's solve each of the given expressions step by step.
1. Correcting the given expression: The first expression [tex]\( \sqrt{\frac{100}{121}} = -10 \)[/tex] is incorrect. The square root of a fraction cannot be negative if it's derived from positive real numbers. The correct square root of [tex]\(\frac{100}{121}\)[/tex] should be:
[tex]\[
\sqrt{\frac{100}{121}} = \sqrt{\frac{10^2}{11^2}} = \frac{10}{11}
\][/tex]
2. Cube root of [tex]\(\frac{27}{512}\)[/tex]:
[tex]\[
\sqrt[3]{\frac{27}{512}} = 0.375
\][/tex]
3. Fourth root of [tex]\(\frac{16}{81}\)[/tex]:
[tex]\[
\sqrt[4]{\frac{16}{81}} = 0.6666666666666666
\][/tex]
4. Square root of [tex]\(\frac{169}{225}\)[/tex]:
[tex]\[
\sqrt{\frac{169}{225}} = 0.8666666666666666
\][/tex]
5. Sixth root of [tex]\(\frac{1}{64}\)[/tex]:
[tex]\[
\sqrt[6]{\frac{1}{64}} = 0.5
\][/tex]
So, the detailed step-by-step solution provides the following results for each expression:
[tex]\[
\sqrt[3]{\frac{27}{512}} = 0.375
\][/tex]
[tex]\[
\sqrt[4]{\frac{16}{81}} = 0.6666666666666666
\][/tex]
[tex]\[
\sqrt{\frac{169}{225}} = 0.8666666666666666
\][/tex]
[tex]\[
\sqrt[6]{\frac{1}{64}} = 0.5
\][/tex]
