SOLUTION: Simplify with positive exponents: {{{((a^3*b^(-2))^(-2)a^2b/(a^(-1)b))^(-2)}}}

Algebra ->  Exponents -> SOLUTION: Simplify with positive exponents: {{{((a^3*b^(-2))^(-2)a^2b/(a^(-1)b))^(-2)}}}      Log On


   



Question 169594: Simplify with positive exponents:
%28%28a%5E3%2Ab%5E%28-2%29%29%5E%28-2%29a%5E2b%2F%28a%5E%28-1%29b%29%29%5E%28-2%29

Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!

%28%28a%5E3%2Ab%5E%28-2%29%29%5E%28-2%29a%5E2b%2F%28a%5E%28-1%29b%29%29%5E%28-2%29

Put 1 exponents so everything will show an exponent:

%28%28a%5E3%2Ab%5E%28-2%29%29%5E%28-2%29a%5E2b%5E1%2F%28a%5E%28-1%29b%5E1%29%29%5E%28-2%29


First do this part: %28a%5E3%2Ab%5E%28-2%29%29%5E%28-2%29
,
Then replace %28a%5E3%2Ab%5E%28-2%29%29%5E%28-2%29 by a%5E%28-6%29b%5E4 

%28a%5E%28-6%29b%5E4a%5E2b%5E1%2F%28a%5E%28-1%29b%5E1%29%29%5E%28-2%29

Inside the parentheses add exponents of a and of b:

%28%28a%5E%28-4%29b%5E5%29%2F%28a%5E%28-1%29b%5E1%29%29%5E%28-2%29

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



%28a%5E8b%5E%28-10%29%29%2F%28a%5E2b%5E%28-2%29%29

Move b%5E%28-10%29 from numerator to denominator and
change the sign of the exponent, and write b%5E10
on the bottom.
Likewise, move b%5E%28-2%29 from denominator to numerator and
change the sign of the exponent, and write b%5E2
on the top. 

%28a%5E8b%5E2%29%2F%28a%5E2b%5E10%29

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%5E%288-2%29 or a%5E6
and place this on the top since the exponent 8 was
larger than the exponent 2 and a%5E8 was on the top.
We eliminate a%5E2 from the bottom:

%28a%5E6b%5E2%29%2F%28b%5E10%29

Finally we subtract exponents of b, b%5E%2810-2%29 or b%5E8
and place this on the bottom since the exponent 10 was
larger than the exponent 2 and b%5E10 was on the bottom.
We eliminate b%5E2 from the top:

%28a%5E6%29%2F%28b%5E8%29

Edwin