Question

which of the following $x$ values are solutions to the exponential equation?
$-dfrac{(-9)^{(x-6)}}{2} = -dfrac{1}{162}$
show your work here
$\bigcirc$ 3
$\bigcirc$ 5
$\bigcirc$ 4
$\bigcirc$ 6

Explanation:

Step1: Eliminate negative signs

Multiply both sides by $-1$:
$$\frac{(-9)^{(x-6)}}{2} = \frac{1}{162}$$

Step2: Isolate the exponential term

Multiply both sides by $2$:
$$(-9)^{(x-6)} = \frac{2}{162} = \frac{1}{81}$$

Step3: Rewrite bases as powers of 9

Express both sides with base $9$:
$$(-9)^{(x-6)} = 9^{-2}$$
Note that $(-9)^{(x-6)} = 9^{(x-6)}$ when $x-6$ is even, since negative base raised to even power is positive.

Step4: Solve for exponent equality

Set exponents equal (since bases match and result is positive):
$$x-6 = -2$$
$$x = 6 - 2 = 4$$
Verify: $(-9)^{4-6}=(-9)^{-2}=\frac{1}{(-9)^2}=\frac{1}{81}$, which matches $\frac{1}{81}$.

Answer:

4