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Find the value of n, where n is an integer and 2^{n–5} × 6^{2n–4} = `1/(12^4 xx 2)`.

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#### Solution

We have, 2^{n–5} × 6^{2n–4} = `1/(12^4 xx 2)`

⇒ `2^n/2^5 xx 6^(2n)/6^4 = 1/(12^4 xx 2)` ......`[∵ a^(m-n) = a^m/a^n]`

⇒ `(2^n xx 6^(2n))/(2^5 xx 6^4) = 1/((2 xx 6)^4 xx 2)` ......[∵ 12 = 6 × 2]

⇒ 2^{n} × (6^{2})^{n} = `(2^5 xx 6^4)/(2^4 xx 6^4 xx 2)` ......[By cross-multiplication] [∵ a^{mn} = (a^{m})^{n} and (a × b)^{m} = a^{m} × a^{m+n}]

⇒ 2^{n} × 36^{n} = `(2^5 xx 6^4)/(2^5 xx 6^4)` ......[∵ a^{m} × a^{n} = a^{m+n}]

⇒ 2^{n} × 36^{n} = 1

⇒ (2 × 36)^{n} = 1 ......[∵ a^{m} × b^{m} = (ab)^{m}]

⇒ (72)^{n} = (72)^{0} ......[∵ a^{0} = 1]

∴ n = 0 ......[∵ If a^{m} = a^{n} ⇒ m = n]

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