Question

How do you multiply `(\frac { 45} { 3t } ) ^ { - 2} \cdot ( \frac { 2s } { 6t } ) ^ { 2}`?

Answer 1

See a solution process below:

Explanation:

First, use these rules of exponents to expand each term in parenthesis:

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

`(45/(3t))^-2 * ((2s)/(6t))^2 =>`

`(45^color(red)(1)/(3^color(red)(1)t^color(red)(1)))^color(blue)(-2) * ((2^color(red)(1)s^color(red)(1))/(6^color(red)(1)t^color(red)(1)))^color(blue)(2) =>`

`(45^(color(red)(1) * color(blue)(-2))/(3^(color(red)(1) * color(blue)(-2))t^(color(red)(1) * color(blue)(-2)))) * ((2^(color(red)(1) * color(blue)(2))s^(color(red)(1) * color(blue)(2)))/(6^(color(red)(1) * color(blue)(2))t^(color(red)(1) * color(blue)(2)))) =>`

`45^-2/(3^-2t^-2) * (2^2s^2)/(6^2t^2) =>`

`45^-2/(3^-2t^-2) * (4s^2)/(36t^2)`

We can now use these rules of exponents to rewrite the term on the left:

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

`45^color(red)(-2)/(3^color(red)(-2)t^color(red)(-2)) * (4s^2)/(36t^2) =>`

`(3^color(red)(- -2)t^color(red)(- -2))/45^color(red)(- -2) * (4s^2)/(36t^2) =>`

`(3^color(red)(2)t^color(red)(2))/45^color(red)(2) * (4s^2)/(36t^2) =>`

`(9t^2)/2025 * (4s^2)/(36t^2) =>`

`(36s^2t^2)/(2025 * 36t^2) =>`

`(color(red)(cancel(color(black)(36)))s^2color(green)(cancel(color(black)(t^2))))/(2025 * color(red)(cancel(color(black)(36)))color(green)(cancel(color(black)(t^2)))) =>`

`s^2/2025`