Question

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solve. write your answer as an integer or a fraction in simplest form.
$25 = \left(\frac{1}{625}\
ight)^{4x + 2}$
$x = \square$

Explanation:

Step1: Express bases as powers of 5

$25 = 5^2$, $\frac{1}{625}=5^{-4}$
So the equation becomes:
$5^2 = (5^{-4})^{4x+2}$

Step2: Simplify the right-hand side

Use exponent rule $(a^m)^n=a^{m \cdot n}$:
$5^2 = 5^{-4(4x+2)}$

Step3: Set exponents equal

Since bases are equal, exponents are equal:
$2 = -4(4x+2)$

Step4: Solve for the linear term

Divide both sides by -4:
$\frac{2}{-4} = 4x+2$
$\frac{-1}{2} = 4x+2$

Step5: Isolate the variable term

Subtract 2 from both sides:
$\frac{-1}{2} - 2 = 4x$
$\frac{-1}{2} - \frac{4}{2} = 4x$
$\frac{-5}{2} = 4x$

Step6: Solve for x

Divide both sides by 4:
$x = \frac{-5}{2 \times 4}$
$x = \frac{-5}{8}$

Answer:

$x = -\frac{5}{8}$