$\displaystyle a^2 x a^{1/2}= a^{2+ 1/2}= a^{5/2}$, not $\displaystyle a^{5/8}$.

Was that a typo?

3. 3b^1/2 x 2b^1/2 x 6b^-1

=36b

$\displaystyle b^{1/2}b^{1/2}b^{-1}= b^{1/2+ 1/2- 1}= b^{1- 1}= b^0= 1$

4.

__7pq__
14p^2q^-2

=0.5p^-1q^3

correct.

5.

__(3xy)^2__

4xy^1/2

=2.25xy^1.5

Correct.

6.

__√s^2t^4 x 3√8s^3t^9__
st

=8^1/3s^2t^4

It's not clear where the square roots end. Use parentheses.

7. 2a^1/2b^1/2c^-1/2 x 3a^3/2b^1/2c^3/2

=6a^4/2bc

$\displaystyle a^{1/2}a^{3/2}= a^{1/2+ 3/2}= a^2$

8. (a^2b^3)^2 x a^-2b^-1

=a^2b^8

$\displaystyle (b^3)^2(b^{-1})= (b^6)(b^{-1})= b^{6- 1}$

( $\displaystyle (x^a)^b= x^{ab}$, NOT $\displaystyle x^{a^b}$)

9. (5ab)^-2 x 15a^2b^3

= 0.6ab