2a^-2 / (8a)^-1
a^-2 = 1/a^2
1 / (8a)^-1 = 1 / (1/8a) = 8a
therefore 2a^-2 / (8a)^-1 = 2 * 2 * 8a / a^2 = 16a/a^2 = 16/a
you can confirm you did this correctly by taking any random value of a and placing it in your calculator and evaluating the original expression and the final expression.
i used a = 15.
in my calculator, i entered:
2a^-2 / (8a)^-1
(2 * 15^-2) / (8*15)^-1 and got the result equal to 1.066666667.
i then entered 16 / 15 and got 1.066666667.
the answers are the same, meaning the original expression is equivalent to the final expreswsion meaning i simplified the expression correctly.
if you explicitly show the operations, it might be easier to analyze.
2a^-2 / (8a)^-1 is the same as (2 * a^-2) / (8a)^-1
this is equivalent to (2^1 * a^-2) / (8a)^-1
a simple trick is, that if you have a variable or numkber that is raised to the negative exponent and .......
if it is in the numerator, place it in the denominator and make the exponent positive.
if it is in the denominator, place it in the numerator and make the exponent negative.
this is easier to see when you manually create the fraction as shown below:
(2^1 * a^-2) / (8a)^-1 becomes:
2^1 * a^-2
--------------------
(8a)^-1
put the a^-2 in the denominator and make it a^2.
put the (8a)^-1 in the numerator and make it (8a)^2.
you will get:
2^1 * (8a)^1
--------------------
a^2
simplify toget:
2 * 8a
-----------------
a^2
simplify further to get:
16a
------------
a^2
simplify further to get:
16
--------
a
other things you should take into consideration.
x^n / x^m = x^(n-m) = 1/x(m-n)
x^n * x^m = x^(n+m)
one of these is why 16a/a^2 is equal to 16 * a^(1-2) which is equal to 16 * a^-1 which is equal to 16/a.
also, 16a / a^2 is equal to 16 / (a^2 - a^1) which is equal to 16 / a^1 which is equal to 16/a.
here a very good reference on exponent arithmetic.
https://https://www.purplemath.com/modules/exponent.htm
there are 5 web pages involved, so be sure to go through each one to get the complete picture provided.