1. **State the problem:** Simplify the expression $$\frac{\sqrt[5]{\sqrt[10]{.}} \sqrt[10]{a^2 b}}{\sqrt[10]{10}}$$ and determine which of the options (a) to (d) it equals. 2. **Rewrite the radicals using fractional exponents:** - The term $\sqrt[5]{\sqrt[10]{.}}$ is ambiguous as written, but assuming it means $\sqrt[5]{\sqrt[10]{10}}$, we rewrite: $$\sqrt[10]{10} = 10^{\frac{1}{10}}$$ Then $$\sqrt[5]{\sqrt[10]{10}} = \left(10^{\frac{1}{10}}\right)^{\frac{1}{5}} = 10^{\frac{1}{10} \times \frac{1}{5}} = 10^{\frac{1}{50}}$$ 3. **Rewrite the numerator:** $$\sqrt[5]{\sqrt[10]{10}} \sqrt[10]{a^2 b} = 10^{\frac{1}{50}} \times (a^2 b)^{\frac{1}{10}} = 10^{\frac{1}{50}} a^{\frac{2}{10}} b^{\frac{1}{10}} = 10^{\frac{1}{50}} a^{\frac{1}{5}} b^{\frac{1}{10}}$$ 4. **Rewrite the denominator:** $$\sqrt[10]{10} = 10^{\frac{1}{10}}$$ 5. **Combine numerator and denominator:** $$\frac{10^{\frac{1}{50}} a^{\frac{1}{5}} b^{\frac{1}{10}}}{10^{\frac{1}{10}}} = a^{\frac{1}{5}} b^{\frac{1}{10}} \times 10^{\frac{1}{50} - \frac{1}{10}}$$ Calculate the exponent of 10: $$\frac{1}{50} - \frac{1}{10} = \frac{1}{50} - \frac{5}{50} = -\frac{4}{50} = -\frac{2}{25}$$ So $$= a^{\frac{1}{5}} b^{\frac{1}{10}} 10^{-\frac{2}{25}} = \frac{a^{\frac{1}{5}} b^{\frac{1}{10}}}{10^{\frac{2}{25}}}$$ 6. **Check the options:** - (a) $\sqrt[10]{10 a^2 b} = (10 a^2 b)^{\frac{1}{10}} = 10^{\frac{1}{10}} a^{\frac{2}{10}} b^{\frac{1}{10}} = 10^{\frac{1}{10}} a^{\frac{1}{5}} b^{\frac{1}{10}}$ - (b) $a \sqrt[10]{b} = a b^{\frac{1}{10}} = a^{1} b^{\frac{1}{10}}$ - (c) $\sqrt[10]{a b} = (a b)^{\frac{1}{10}} = a^{\frac{1}{10}} b^{\frac{1}{10}}$ - (d) $\sqrt[5]{a} \sqrt[10]{b} = a^{\frac{1}{5}} b^{\frac{1}{10}}$ 7. **Compare with our simplified expression:** Our expression is $\frac{a^{\frac{1}{5}} b^{\frac{1}{10}}}{10^{\frac{2}{25}}}$, which is not exactly equal to any option because of the denominator $10^{\frac{2}{25}}$. However, if the initial term $\sqrt[5]{\sqrt[10]{.}}$ was intended to be $\sqrt[5]{1} = 1$, then the numerator would be $\sqrt[10]{a^2 b} = a^{\frac{1}{5}} b^{\frac{1}{10}}$ and denominator $\sqrt[10]{10} = 10^{\frac{1}{10}}$, so the expression would be $$\frac{a^{\frac{1}{5}} b^{\frac{1}{10}}}{10^{\frac{1}{10}}}$$ which does not match any option exactly either. Given the options and the problem context, the closest match ignoring the factor involving 10 is option (d): $$\sqrt[5]{a} \sqrt[10]{b} = a^{\frac{1}{5}} b^{\frac{1}{10}}$$ **Final answer:** (d) $\sqrt[5]{a} \sqrt[10]{b}$