Question

How do you simplify `4a^(3)b^(2) * 3a^(-4)b^(-3)`?

Answer 1

The result is `12/(ab)`.

Explanation:

You can switch around the order of the multiplication. I color-coded the like terms so they are easier to see:

`color(white)=color(red)4color(blue)(a^3)color(green)(b^2)*color(red)color(blue)(3a^-4)color(green)(b^-3)`

`=color(red)4*color(blue)(a^3)*color(green)(b^2)*color(red)3*color(blue)(a^-4)*color(green)(b^-3)`

`=color(red)4*color(red)3*color(blue)(a^3)*color(blue)(a^-4)*color(green)(b^2)*color(green)(b^-3)`

`=color(red)12*color(blue)(a^(3+(-4)))*color(green)(b^(2+(-3)))`

`=color(red)12*color(blue)(a^(3-4))*color(green)(b^(2-3))`

`=color(red)12*color(blue)(a^-1)*color(green)(b^-1)`

`=color(red)12*color(blue)(1/a)*color(green)(1/b)`

`=color(red)12*1/(color(blue)acolor(green)b)`

`=color(red)12/(color(blue)acolor(green)b)`

That is the result.

Answer 2

`12 a^-1 b^-1`   same as   `(12)/(a b)`

Explanation:

There are a couple of ways to solve this problem.

1) Using the laws of exponents, you multiply them by adding.

So you can just multiply like bases by addition of their exponents.

Simplify

`4 a^3 b^2 ⋅ 3  a^-4 b^-3`

1) This is just one long multiplication problem anyway, so you can rearrange the factors for your own convenience.

`(4)(3)   (a^3)(a^-4)   (b^2)(b^-3)`

2) Now you can multiply by adding the exponents of like bases

`12  a^-1  b^-1` `larr` answer

`color(white)(mmmmmmmm)`――――――――――――

2) You can write the problem as a fraction and reduce it to lowest terms

You can get rid of the negative exponents by writing them in the denominator and changing their signs.

Simplify

`4 a^3 b^2 ⋅ 3 a^-4  b^-3`

1) Write any factors with negative exponents in the denominator and change the sign of the exponent

`((4*3 a^3 b^2))/(a^4 b^3)`

2) Reduce the fraction to lowest terms

`(12)/(a b)` `larr` same answer