A and b are numbers

Argile1845

New member
A and b are numbers and a^3=5 and b^3=4.
Find the value of (ab^2) ^-3

I need help starting this. This one is stumping me but I know it's simple I'm just looking at it all wrong.

Subhotosh Khan

Super Moderator
Staff member
A and b are numbers and a^3=5 and b^3=4.
Find the value of (ab^2) ^-3

I need help starting this. This one is stumping me but I know it's simple I'm just looking at it all wrong.
Use:

(xn)m = x(n*m)

and simplify (ab^2) ^(-3)

Dr.Peterson

Elite Member
A and b are numbers and a^3=5 and b^3=4.
Find the value of (ab^2) ^-3

I need help starting this. This one is stumping me but I know it's simple I'm just looking at it all wrong.
You can first apply the fact that (xy)^n = x^n y^n.

Then you can apply the fact that (x^m)^n = x^(mn).

Are you familiar with these?

There will be more to do, but that is a start.

Give it a try and show some work, so we see where you stand.

Argile1845

New member
Well so far I have ab^-6
I know that a^3=5 and b^3=4 so if I split the ^-6 up I'd have -5*-5*-4*-4=400?

Subhotosh Khan

Super Moderator
Staff member
Well so far I have ab^-6
I know that a^3=5 and b^3=4 so if I split the ^-6 up I'd have -5*-5*-4*-4=400?
It should be (ab)^(-6)

(ab)^(-6) = [(a * b)^(3)]^(-2) = [(a^3 * b^3)]^(-2) ....... continue

Jomo

Elite Member
I'll do a similar one for you.
a and b are numbers and a^3=3 and b^3=2.
Find the value of (ab^3) ^-3

(ab^4) ^-3 = a-3b-12 = 1/[a3b12] = 1/[3*24]=1/[3*16] = 1/48

Argile1845

New member
I'll do a similar one for you.
a and b are numbers and a^3=3 and b^3=2.
Find the value of (ab^3) ^-3

(ab^4) ^-3 = a-3b-12 = 1/[a3b12] = 1/[3*24]=1/[3*16] = 1/48
Mat I ask why you changed the sign in the step from (ab^3)^-3 to (ab^4) ^-3? Why did the exponent change to a 4th power when it was a 3 to begin with?

Argile1845

New member
It should be (ab)^(-6)

(ab)^(-6) = [(a * b)^(3)]^(-2) = [(a^3 * b^3)]^(-2) ....... continue
Ok, so [(a^3 * b^3)]^(-2) the first step is do is the inner parentheses- [(5*4)]^(-2) = (20)^(-2) =
Mat I ask why you changed the sign in the step from (ab^3)^-3 to (ab^4) ^-3? Why did the exponent change to a 4th power when it was a 3 to begin with?
Nevermind, I noticed my mistake

Subhotosh Khan

Super Moderator
Staff member
Ok, so [(a^3 * b^3)]^(-2) the first step is do is the inner parentheses- [(5*4)]^(-2) = (20)^(-2) = CORRECT

Argile1845

New member
Aaahhh, do you know how happy that makes me?! Whew. Thank you so much. Learning on my own and it's so difficult. Thanks again! Sorry for ally questions but I have no one else.

Argile1845

New member
I'm just concerned about one thing...
On the original question it had (ab^2)^-3... But in the question above that I solved I was multiplying the a and the b by the ^3power before multiplying it by a ^ -2 power.. Can I really switch those places?

Jomo

Elite Member
Mat I ask why you changed the sign in the step from (ab^3)^-3 to (ab^4) ^-3? Why did the exponent change to a 4th power when it was a 3 to begin with?
Things like that happen when you use a similar problem. A similar problem is different from the original one.

Dr.Peterson

Elite Member
I'll do a similar one for you.
a and b are numbers and a^3=3 and b^3=2.
Find the value of (ab^3) ^-3

(ab^4) ^-3
= a-3b-12 = 1/[a3b12] = 1/[3*24]=1/[3*16] = 1/48
You do realize you made a typo in your own problem, right?

Argile1845

New member
That's what I thought and is what I was asking.

Argile1845

New member
You can first apply the fact that (xy)^n = x^n y^n.

Then you can apply the fact that (x^m)^n = x^(mn).

Are you familiar with these?

There will be more to do, but that is a start.

Give it a try and show some work, so we see where you stand.
So would this be [(a(^2 * ^-3) * b(^2 * ^-3)]
Which would equal (a^-6 * b^-6) correct?
I know a^3=5 so I want to simplify this and say (-5*-5*-4*-4) = 400... I still get 400.
What am I doing wrong?

Dr.Peterson

Elite Member
So would this be [(a(^2 * ^-3) * b(^2 * ^-3)]
Which would equal (a^-6 * b^-6) correct?
I know a^3=5 so I want to simplify this and say (-5*-5*-4*-4) = 400... I still get 400.
What am I doing wrong?
What you wrote there is meaningless; I suppose you meant a^(2*-3) b^(2*-3). But if you started with (ab^2)^-3 from your problem, that is wrong. Do you see where?

Also, if a^3 = 5, then a^-6 is not -5*-5. In fact, a^-3 is not -5. Negative exponents don't change the sign of the result; they product a reciprocal: a^-n = 1/a^n, not -a^n. That is probably a big part of your difficulty.

Argile1845

New member
This helps a lot, thank you. I realize that now, how did I forget that?!
I am sorry about the typing. I am new at having to type it out and I can get mess it up. Thank you for being patient with me.

Argile1845

New member
That would be 1/5*4*4= 1/80

Dr.Peterson

Elite Member
A and b are numbers and a^3=5 and b^3=4.
Find the value of (ab^2) ^-3
That would be 1/5*4*4= 1/80
You mean 1/(5*4*4) = 1/80.

I think you've got it: $$\displaystyle (ab^2)^{-3}=a^{-3}b^{-6}=\frac{1}{a^3b^6}=\frac{1}{a^3(b^3)^2}=\frac{1}{5(4)^2}=\frac{1}{80}$$