Questions on Algebra: Exponents and operations on exponents answered by real tutors!

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Question 169594: Simplify with positive exponents:
((a^3*b^(-2))^(-2)a^2b/(a^(-1)b))^(-2)
: Simplify with positive exponents:
((a^3*b^(-2))^(-2)a^2b/(a^(-1)b))^(-2)

Answer by Edwin McCravy(2190) About Me  (Show Source):
You can put this solution on YOUR website!

((a^3*b^(-2))^(-2)a^2b/(a^(-1)b))^(-2)

Put 1 exponents so everything will show an exponent:

((a^3*b^(-2))^(-2)a^2b^1/(a^(-1)b^1))^(-2)


First do this part: (a^3*b^(-2))^(-2)
(a^3*b^(-2))^(-2)= a^(3*(-2))*b^((-2)(-2))=a^(-6)b^4,
Then replace (a^3*b^(-2))^(-2) by a^(-6)b^4 

(a^(-6)b^4a^2b^1/(a^(-1)b^1))^(-2)

Inside the parentheses add exponents of a and of b:

((a^(-4)b^5)/(a^(-1)b^1))^(-2)

Now we're ready to remove the parentheses by multiplying
every inner exponent top and bottom by the outer
exponent -2:

(a^((-4)(-2))b^(5(-2)))/(a^((-1)(-2))b^(1(-2)))

(a^8b^(-10))/(a^2b^(-2))

Move b^(-10) from numerator to denominator and
change the sign of the exponent, and write b^10
on the bottom.
Likewise, move b^(-2) from denominator to numerator and
change the sign of the exponent, and write b^2
on the top. 

(a^8b^2)/(a^2b^10)

Subtract exponents, larger minus smaller, and place the
result in the numerator or denominator, depending on
which one had the larger exponent before.

So we subtract exponents of a, a^(8-2) or a^6
and place this on the top since the exponent 8 was
larger than the exponent 2 and a^8 was on the top.
We eliminate a^2 from the bottom:

(a^6b^2)/(b^10)

Finally we subtract exponents of b, b^(10-2) or b^8
and place this on the bottom since the exponent 10 was
larger than the exponent 2 and b^10 was on the bottom.
We eliminate b^2 from the top:

(a^6)/(b^8)

Edwin