Question

question
solve for x:
\\(1296^{3x - 3} = 216^{3x - 5}\\)

Explanation:

Step1: Express bases as powers of 6

$1296 = 6^4$, $216 = 6^3$
Substitute into equation:
$$(6^4)^{3x-3} = (6^3)^{3x-5}$$

Step2: Simplify exponents using power rule

Power rule: $(a^m)^n = a^{m \cdot n}$
$$6^{4(3x-3)} = 6^{3(3x-5)}$$

Step3: Set exponents equal (same base)

Since bases are equal, exponents must be equal:
$$4(3x-3) = 3(3x-5)$$

Step4: Expand both sides of equation

$$12x - 12 = 9x - 15$$

Step5: Isolate x terms on left

Subtract $9x$ from both sides:
$$12x - 9x - 12 = -15$$
$$3x - 12 = -15$$

Step6: Isolate x constant on right

Add 12 to both sides:
$$3x = -15 + 12$$
$$3x = -3$$

Step7: Solve for x

Divide both sides by 3:
$$x = \frac{-3}{3}$$

Answer:

$x = -1$