Question

How do you simplify `\frac { 3^ { 2} \cdot 3^ { 3} } { 2^ { 2} \cdot 3^ { 4} }`?

Answer 1

See a couple of process below:

Explanation:

First Process:

We can use this rule of exponents to combine the terms in the numerator:

`x^color(red)(a) xx x^color(blue)(b) = x^(color(red)(a) + color(blue)(b))`

`(3^color(red)(2) * 3^color(blue)(3))/(2^2 * 3^4) =>`

`3^(color(red)(2) + color(blue)(3))/(2^2 * 3^4) =>`

`3^5/(2^2 * 3^4)`

We can now use these rules of exponents to complete the simplification of the `3` terms:

`x^color(red)(a)/x^color(blue)(b) = x^(color(red)(a)-color(blue)(b))` and `a^color(red)(1) = a`

`3^color(red)(5)/(2^2 * 3^color(blue)(4)) =>`

`3^(color(red)(5)-color(blue)(4))/2^2 =>`

`3^color(red)(1)/2^2 =>`

`3/2^2 =>`

`3/4`

Second Process:

`(3^2 * 3^3)/(2^2 * 3^4) => (9 * 27)/(4 * 81) => (3 * 3 * 27)/(4 * 81) => (3 * 81)/(4 * 81) => (3 * color(red)(cancel(color(black)(81))))/(4 * color(red)(cancel(color(black)(81)))) => 3/4`