Reasoning: $\sqrt{a^3}=(a^3)^{\frac{1}{2}}=a^{\frac{3}{2}}$

Reasoning: $a^{\frac{1}{2}}=\sqrt{a}$

Reasoning: $\sqrt{a^2}=a$ and $\sqrt{b^4}=b^2$, therefore $\sqrt{a^2b^4}=ab^2$

Reasoning: $(a^m)^n=a^{mn}$

$(a^3)^{\frac{2}{3}}=a^{3\times\frac{2}{3}}=a^2$

Reasoning: $\sqrt[3]{a^6}=a^{\frac{6}{3}}=a^2$

Reasoning: $\frac{a^m}{a^n}=a^{m-n}$

$\frac{a^{\frac{5}{2}}}{a^{\frac{1}{2}}}=a^{\frac{5}{2}-\frac{1}{2}}=a^{\frac{4}{2}}=a^2$

Reasoning: $a^m\times a^n=a^{m+n}$

$a^{\frac{3}{2}}\times a^{\frac{1}{2}}=a^{\frac{4}{2}}=a^2$

Reasoning: $\sqrt{49}=7$ and $\sqrt{a^2}=a$

Therefore $\sqrt{49a^2}=7a$

Reasoning: Negative index means reciprocal.

$a^{-1}=\frac{1}{a}$

Reasoning: $a^{\frac{1}{3}}\times a^{\frac{2}{3}}=a^{\frac{3}{3}}=a$

Reasoning: $\sqrt[4]{a^8}=a^{\frac{8}{4}}=a^2$

Reasoning: $\frac{a^5}{a^2}=a^{5-2}=a^3$

Reasoning: $a^{\frac{1}{2}}\times a^{\frac{3}{2}}=a^{\frac{4}{2}}=a^2$

Reasoning: $\frac{1}{\sqrt{a}}=\frac{1}{a^{\frac{1}{2}}}=a^{-\frac{1}{2}}$

Reasoning: $(a^2b^3)^{\frac{1}{2}}=a^{2\times\frac{1}{2}}b^{3\times\frac{1}{2}}=ab^{\frac{3}{2}}$

Reasoning: $\sqrt{a^4}=a^2$ and $\sqrt{b^2}=b$

Therefore $\sqrt{a^4b^2}=a^2b$

Reasoning: $a^{\frac{2}{3}}\div a^{\frac{1}{3}}=a^{\frac{2}{3}-\frac{1}{3}}=a^{\frac{1}{3}}$

Reasoning: $\sqrt[3]{a^3b^6}=a^{\frac{3}{3}}b^{\frac{6}{3}}=ab^2$