(iii) (6(8)^(n+1)+16(2)^(3n-2))/(10(2)^(3n+1)-7(8)^(n))

Simplify terms involving powers: First, simplify the terms involving powers of 8 and 2. (6(8)^(n+1) = 6 * 8 * 8^n = 48 * 8^n, 16(2)^(3n-2) = 16 * 2^(3n-2), 10(2)^(3n+1) = 10 * 2 * 2^(3n) = 20 * 2^(3n), 7(8)^(n) = 7 * 8^n.Rewrite using simplified terms: Rewrite the expression using simplified terms: (48 * 8^n + 16 * 2^(3n-2)) / (20 * 2^(3n) - 7 * 8^n).Recognize and substitute: Recognize that 8^n = (2^3)^n = 2^(3n) and substitute: (48 * 2^(3n) + 16 * 2^(3n-2)) / (20 * 2^(3n) - 7 * 2^(3n)).Factor out common terms: Factor out common terms in the numerator and denominator: 2^(3n-2) * (48 * 4 + 16) / 2^(3n) * (20 - 7), = 2^(3n-2) * (192 + 16) / 2^(3n) * 13, = 2^(3n-2) * 208 / 2^(3n) * 13.Reduce powers of 2: Simplify by reducing the powers of 2: 2^(3n-2) / 2^(3n) = 2^(-2), = 2^(-2) * 208 / 13, = 208 / (13 * 4), = 208 / 52.Perform division: Perform the division: 208 / 52 = 4.

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Full solution

Q. (iii)

\frac{6(8)^{n+1}+16(2)^{3 n-2}}{10(2)^{3 n+1}-7(8)^{n}}

Simplify terms involving powers: First, simplify the terms involving powers of $8$ and $2$ .
$(6(8)^{(n+1)} = 6 \times 8 \times 8^n = 48 \times 8^n,$
$16(2)^{(3n-2)} = 16 \times 2^{(3n-2)},$
$10(2)^{(3n+1)} = 10 \times 2 \times 2^{(3n)} = 20 \times 2^{(3n)},$
$7(8)^{(n)} = 7 \times 8^n.$
Rewrite using simplified terms: Rewrite the expression using simplified terms: $(48 \cdot 8^n + 16 \cdot 2^{3n-2}) / (20 \cdot 2^{3n} - 7 \cdot 8^n)$ .
Recognize and substitute: Recognize that $8^n = (2^3)^n = 2^{3n}$ and substitute: $\newline$ $(48 \cdot 2^{3n} + 16 \cdot 2^{3n-2}) / (20 \cdot 2^{3n} - 7 \cdot 2^{3n})$ .
Factor out common terms: Factor out common terms in the numerator and denominator: $\newline$ $2^{3n-2} \times (48 \times 4 + 16) / 2^{3n} \times (20 - 7)$ , $\newline$ $= 2^{3n-2} \times (192 + 16) / 2^{3n} \times 13$ , $\newline$ $= 2^{3n-2} \times 208 / 2^{3n} \times 13$ .
Reduce powers of $2$ : Simplify by reducing the powers of $2$ : $\newline$ $2^{3n-2} / 2^{3n} = 2^{-2}$ , $\newline$ $= 2^{-2} \times 208 / 13$ , $\newline$ $= 208 / (13 \times 4)$ , $\newline$ $= 208 / 52$ .
Perform division: Perform the division: $208 / 52 = 4$ .