Dividing with indices resulting in incorrect sign

[Dickie]

Homework Statement

Simplify [(-2t)^3] / [(-4t)^2] and evaluate (-2/3)^-2

Relevant Equations

1/m^n = m^-n

I am able to simplify/evaluate the above equations correctly, however I end up with an incorrect sign for each answer (i.e positive when it should be negative) and I can't see where the error is. I feel I am clearly missing something but having checked my working including with a calculator for the basic arithmetic (to check the signs) I am none the wiser as to what I am actually getting wrong. Below are my workings:

= [(-2t)^3] / [(-4t)^2]
= [-8t^3] / [-16t^2]
= 1/2t

I am basing this on -8 / -16 = 1/2 and (t^3)/(t^2) = t, although the answer I am provided gives -1/2t which is leading to my confusion. I have also tried beginning with [(-2t)^3)] * [(-4t)^-2] giving [-8t^3] * [-1/16t^-2] however still end up with the same result.

= (-2/3)^-2
= (-2^1 * 3^-1)^-2
= (-2^-2 * 3^2)
= -1/4 * 9
= -9/4

However, again the answer provided is the opposite sign (in this case 9/4).

Looking back, my confusion is to do with a negative divided by a negative vs a negative fraction. So, if -(2/3) = (-2/3) = (-2/-3) [apologies, I can't think of a clearer way of explaining this] then I'd expect:
= -8 / -16
= -(8 / 16) = -8 / 16
= -1/2 which would give me the correct answer for the first question.

In that case I'd expect:
= (-2/3)
= -(2/3) = (-2 / -3)
So:
= (-2^1 * -3^-1)^-2
= -1/4 * -9
= 9/4

Is this then the correct logic I should be using?

 

[SammyS]

Homework Statement:: Simplify [(-2t)^3] / [(-4t)^2] and evaluate (-2/3)^-2
Relevant Equations:: 1/m^n = m^-n

I am able to simplify/evaluate the above equations correctly, however I end up with an incorrect sign for each answer (i.e positive when it should be negative) and I can't see where the error is. I feel I am clearly missing something but having checked my working including with a calculator for the basic arithmetic (to check the signs) I am none the wiser as to what I am actually getting wrong. Below are my workings:

= [(-2t)^3] / [(-4t)^2]
= [-8t^3] / [-16t^2]
= 1/2t

I am basing this on -8 / -16 = 1/2 and (t^3)/(t^2) = t, although the answer I am provided gives -1/2t which is leading to my confusion. I have also tried beginning with [(-2t)^3)] * [(-4t)^-2] giving [-8t^3] * [-1/16t^-2] however still end up with the same result.

= (-2/3)^-2
= (-2^1 * 3^-1)^-2
= (-2^-2 * 3^2)
= -1/4 * 9
= -9/4

What is (−4)² ?

What is (−2)² ?

 

[Dickie]

So, in answering your question I've realized the following:
(-4)^2 = 16
-4^2 = -16

This seems to be the source of my confusion for the first question, so thank you. This also seems to apply for the second question where I think I should have used:

(-2)^-2 = 1/4

Thanks again - I thought it would be something simple I'd missed!

 

[SammyS]

So, in answering your question I've realized the following:
(-4)^2 = 16
-4^2 = -16

This seems to be the source of my confusion for the first question, so thank you. This also seems to apply for the second question ...

Yes.
Remember the fundamentals.

(−2t)^2 is shorthand for (−2t)×(−2t) , which is 4×t^2

 

[Dickie]

Thanks again, I always find it's the basics which trip me up.

I'm returning to study in my 30s for an MSc and have been given these questions as part of some pre-course work to do to refresh my maths - despite having done well at maths in the past, it turns out there are more than a few gaps in my knowledge now!