Question

\[(x ^{-5} \div y ^{-8})^{4}\]

I want to have a detailed explanation of the solution.
Been struggling to understand the rules that apply on each step. I am pretty thorough with the product rules and everything else but I have hard time with the ones where the base is different.
Any sort of help is appreciated here!

Answer 1

the equation can also be written as \[\frac{(x^{-5})^4}{(y ^{-8})^4}\]

Answer 2

we have \[(n^a)^b = n^{a \times b}\]

try it both for the numerator and denominator

Answer 3

ok

Answer 4

so what I get confused with is that how should I reciprocate? One of the rules I read was to reciprocate the values when there is a negative sign.

So two questions really...
1) Do I need to reciprocate? If yes, then Will the reciprocation invert the sign outside the bracket or just the sign of the denominator?
2) How is the reciprocation done step-by-step?

Answer 5

yes that was true, if any values that have a negative exponent will be reciprocate, in this case, you can either reciprocate it first, or multiply it first by the exponents and then reciprocate.

Answer 6

\[x ^{-20}\div y ^{-32}\]

Answer 7

yes, and then get the reciprocal of the numerator and denominator

Answer 8

what would be the answer @sunnymsus ?

Answer 9

k...would it be \[y ^{32} \div x ^{20}\]

Answer 10

yes it is. :)

Answer 11

o nice...that was easy. let me try another problem like that.

Answer 12

another thing...what if the numerator was positive and the denominator was negative?

Answer 13

if you always practice, it would be a lot easier for you.

Answer 14

yes...i have been trying since yesterday...although i have become better at it but it's still not perfect. hoping with practice i would be able to crack it.

Answer 15

Good job! :)

Answer 16

what if the numerator was positive and the denominator was negative?

Answer 17

will we keep the sign of the numerator as positive even after reciprocation?

Answer 18

the site where I am practicing says the signs are off. It says it is incorrect.

Answer 19

if the numerator has positive exponent and the denominator has negative, only reciprocate the denominator

@sunnymsus what's the problem?

Answer 20

if you pay attention to the answer I had pasted above, I changed the sign's of both numerator and denominator, whereas I think I should have done that only for the denominator? Is that correct?

Answer 21

the answer above is correct. since both have negative exponents we will take their reciprocals.