1. The problem is to simplify the expressions involving fractional exponents. 2. Recall the rule for fractional exponents: $$a^{m/n} = \sqrt[n]{a^m} = \left(\sqrt[n]{a}\right)^m$$ where $m$ is the numerator and $n$ is the denominator of the fraction. 3. Simplify each expression step-by-step: - For $16^{1/2}$: $$16^{1/2} = \sqrt{16} = 4$$ - For $0.04^{1/2}$: $$0.04^{1/2} = \sqrt{0.04} = 0.2$$ - For $16^{3/2}$: $$16^{3/2} = \left(16^{1/2}\right)^3 = 4^3 = 64$$ - For $\left(\frac{64}{125}\right)^{2/3}$: $$\left(\frac{64}{125}\right)^{2/3} = \left(\sqrt[3]{\frac{64}{125}}\right)^2 = \left(\frac{\sqrt[3]{64}}{\sqrt[3]{125}}\right)^2 = \left(\frac{4}{5}\right)^2 = \frac{16}{25}$$ - For $\left(\frac{36}{169}\right)^{-1/2}$: $$\left(\frac{36}{169}\right)^{-1/2} = \frac{1}{\left(\frac{36}{169}\right)^{1/2}} = \frac{1}{\sqrt{\frac{36}{169}}} = \frac{1}{\frac{6}{13}} = \frac{13}{6}$$ 4. Final simplified answers: - $16^{1/2} = 4$ - $0.04^{1/2} = 0.2$ - $16^{3/2} = 64$ - $\left(\frac{64}{125}\right)^{2/3} = \frac{16}{25}$ - $\left(\frac{36}{169}\right)^{-1/2} = \frac{13}{6}$