Simplify the following:
3n×9n+1÷(3n−1×9n−1)3^n \times 9^{n + 1} ÷ (3^{n - 1} \times 9^{n - 1})3n×9n+1÷(3n−1×9n−1)
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Given,
⇒3n×9n+1÷(3n−1×9n−1)=3n×9n+1×13n−1×9n−1=3n×9n.913n.3−1×9n.9−1=93−1×9−1=9×3×9=243.\Rightarrow 3^n \times 9^{n + 1} ÷ (3^{n - 1} \times 9^{n - 1}) \\[1em] = 3^n \times 9^{n + 1} \times \dfrac{1}{3^{n - 1} \times 9^{n - 1}} \\[1em] = \dfrac{3^n \times 9^n.9^1}{3^n.3^{-1} \times 9^n.9^{-1}} \\[1em] = \dfrac{9}{3^{-1} \times 9^{-1}} \\[1em] = 9 \times 3 \times 9 \\[1em] = 243.⇒3n×9n+1÷(3n−1×9n−1)=3n×9n+1×3n−1×9n−11=3n.3−1×9n.9−13n×9n.91=3−1×9−19=9×3×9=243.
Hence, 3n×9n+1÷(3n−1×9n−1)3^n \times 9^{n + 1} ÷ (3^{n - 1} \times 9^{n - 1})3n×9n+1÷(3n−1×9n−1) = 243.
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