Question

If `(5^a * 125^a + 625^a)/(25^-b * 8^(1/3)) = 125`, show that 4a + 2b = 3.

Answer 1

Rewrite everything in base 5:

• 125 = 5³, 625 = 5⁴, 25 = 5², 8^1/3 = 2
• 125^a = 5^3a, 625^a = 5^4a

Then

`(5^a * 125^a + 625^a)/(25^-b * 8^(1//3))`

= `(5^a * 5^(3a) + 5^(4a))/((5^2)^-b * 2)`

= `(5^(4a) + 5^(4a))/(2 * 5^(-2b))`

= `(2 * 5^(4a))/(2 * 5^(-2b))`

= `5^(4a) * 5^(2b)`

= `5^(4a + 2b)`

Given the value is 125 = 5³, we get

`5^(4a + 2b) = 5^3`

Hence 4a + 2b = 3.