Question

solve the equation \\(\left(\frac{17^9}{17^n}\
ight)^2 = (17^3)^4\\) for \\(n\\).
\\(n = -6\\)
\\(\left(\frac{17^9}{17^n}\
ight)^2 = (17^3)^4\\)
\\(\left(\frac{17^{18}}{17^{2n}}\
ight) = (17^3)^4\\)

Explanation:

Step1: Simplify left numerator exponent

$17^{9 \times 2}=17^{18}$

Step2: Simplify left denominator exponent

$17^{n \times 2}=17^{2n}$

Step3: Simplify left side via exponent rule

$17^{18-2n}$

Step4: Simplify right side exponent

$17^{3 \times 4}=17^{12}$

Step5: Set exponents equal (same base)

$18-2n=12$

Step6: Solve for n

$-2n=12-18=-6 \implies n=\frac{-6}{-2}=-6$

Answer:

$n=-6$