Question

select the correct answer.
what is the solution to this equation?
$125^{x - 2} = \left(\frac{1}{25}\
ight)^{3x}$
a. $-\frac{2}{3}$
b. $\frac{2}{9}$
c. $-\frac{9}{2}$
d. $\frac{2}{3}$

Explanation:

Step1: Rewrite bases as powers of 5

$125 = 5^3$, $\frac{1}{25}=5^{-2}$, so:
$$5^{3(x-2)} = 5^{-2(3x)}$$

Step2: Simplify exponents

Expand both sides:
$$5^{3x-6} = 5^{-6x}$$

Step3: Set exponents equal

Since bases are equal, exponents are equal:
$$3x - 6 = -6x$$

Step4: Solve for x

Combine like terms:
$$3x + 6x = 6$$
$$9x = 6$$
$$x = \frac{6}{9} = \frac{2}{3}$$

Answer:

D. $\frac{2}{3}$