Question

In each of the​ following, find all real numbers x that satisfy the given equation. If none satisfy the​ equation, then state the answer as​ "None."

(25/4)^3x = 32/3125​

Answer 1

The solution is
`x = -(5)/(6)`

Explanation:

We solve this question using exponential properties.

We have that:

`((25)/(4))^(3x) = (32)/(3125)`

Simplifying:

`((5^2)/(2^2))^(3x) = (2^5)/(5^5)`

`([(5)/(2)]^(2))^(3x) = ((2)/(5))^(5)`

`((5)/(2))^(6x) = ((2)/(5))^(5)`

Changing the numerator and denominator in the right side:

`((5)/(2))^(6x) = ((5)/(2))^(-5)`

Since both bases are equal:

`6x = -5`

`x = -(5)/(6)`

The solution is
`x = -(5)/(6)`