Question

algebra: concepts and connections - plc solving exponential equations by rewriting the base for what value of ( x ) does ( 3^{2x} = 9^{3x - 4} )?

Explanation:

Step1: Rewrite base 9 as power of 3

$9 = 3^2$, so $9^{3x-4} = (3^2)^{3x-4}$

Step2: Simplify the right-hand side

Using exponent rule $(a^m)^n = a^{mn}$, we get $(3^2)^{3x-4} = 3^{2(3x-4)} = 3^{6x-8}$

Step3: Set exponents equal (same base)

Since $3^{2x} = 3^{6x-8}$, equate exponents: $2x = 6x - 8$

Step4: Solve for x

Subtract $2x$ from both sides: $0 = 4x - 8$
Add 8 to both sides: $8 = 4x$
Divide by 4: $x = \frac{8}{4} = 2$

Answer:

2