How to solve $1/{4a^{-2}}$

Question

Can i write $1/4a^{-2}$ as $4a^2$ ? Or is the right answer to do it like:

$$1/4a^{-2} = 1/4 \cdot 1/a^{-2} = 1/4 \cdot a^2 = a^2/4$$

In the problem there is no parenthesis around $4a$ but assuming there were parenthesis like $1/(4a)^{-2}$ would it be correct to write $4a^2$ ?

Answer 1

Officially, you should parse $1/4a^{-2}$ as $(1/4) \cdot a^{-2}$. The exponential is evaluated first, then the multiplies and divides from left to right. Certainly the $4$ belongs in the denominator. The question is whether the $a^{-2}$ belongs in the denominator. Given that the exponent is negative, I would think it likely that the author meant that. On the other hand, if you see $1/4a^2$ it is likely that the author meant $1/(4a^2)$ instead of $a^2/4$ Parentheses are your friend.

Answer 2

If you have $\dfrac{1}{(4a)^{-2}}$ then it would be $(4a)^2 = 16a^2$. Let me know if you have further questions..