Question

If `5^(10x)=4900` and `2^(sqrty)=25`, what is the value of `((5^((x-1)))^5)/4^(-sqrty)`?

Answer 1

`(5^(x-1))^5/4^(-sqrt(y)) = 14`

Explanation:

Given that:

`5^(10x) = 4900`

`2^(sqrt(y)) = 25`

Then:

`(5^(x-1))^5 = 5^(5(x-1))`

`color(white)((5^(x-1))^5) = 5^(5x-5)`

`color(white)((5^(x-1))^5) = 5^(5x)*5^(-5)`

`color(white)((5^(x-1))^5) = 5^(1/2(10x))*5^(-5)`

`color(white)((5^(x-1))^5) = (5^(10x))^(1/2)*5^(-5)`

`color(white)((5^(x-1))^5) = 4900^(1/2)*5^(-5)`

`color(white)((5^(x-1))^5) = 70*5^(-5)`

`color(white)((5^(x-1))^5) = 14*5^(-4)`

`4^(-sqrt(y)) = (2^2)^(-sqrt(y))`

`color(white)(4^(-sqrt(y))) = 2^(-2sqrt(y))`

`color(white)(4^(-sqrt(y))) = (2^sqrt(y))^(-2)`

`color(white)(4^(-sqrt(y))) = 25^(-2)`

`color(white)(4^(-sqrt(y))) = 5^(-4)`

So:

`(5^(x-1))^5/4^(-sqrt(y)) = (14*color(red)(cancel(color(black)(5^(-4)))))/color(red)(cancel(color(black)(5^(-4)))) = 14`