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

Answer:

The solution is $x = -\frac{5}{6}$

Step-by-step explanation:

We solve this question using exponential properties.

We have that:

$(\frac{25}{4})^{3x} = \frac{32}{3125}$

Simplifying:

$(\frac{5^2}{2^2})^{3x} = \frac{2^5}{5^5}$

$([\frac{5}{2}]^{2})^{3x} = (\frac{2}{5})^{5}$

$(\frac{5}{2})^{6x} = (\frac{2}{5})^{5}$

Changing the numerator and denominator in the right side:

$(\frac{5}{2})^{6x} = (\frac{5}{2})^{-5}$

Since both bases are equal:

$6x = -5$

$x = -\frac{5}{6}$

The solution is $x = -\frac{5}{6}$