To simplify the expression
[tex]\[ \frac{32 \times 3^{n+2} - 16 \times 3^{n+1}}{10 \times 2^{n+2} + 5 \times 2^{n+3}}, \][/tex]
let’s break it into detailed steps:
1. Simplify the numerator:
[tex]\[ 32 \times 3^{n+2} - 16 \times 3^{n+1}. \][/tex]
Notice that both terms have a common factor of [tex]\( 16 \times 3^{n+1} \)[/tex]:
[tex]\[ 32 \times 3^{n+2} = 32 \times 3^n \times 3^2 = 32 \times 3^2 \times 3^n = 288 \times 3^n. \][/tex]
[tex]\[ 16 \times 3^{n+1} = 16 \times 3^n \times 3 = 48 \times 3^n. \][/tex]
So, the numerator becomes:
[tex]\[ 288 \times 3^n - 48 \times 3^n.
= (288 - 48) \times 3^n
= 240 \times 3^n \][/tex]
2. Simplify the denominator:
[tex]\[ 10 \times 2^{n+2} + 5 \times 2^{n+3}. \][/tex]
Notice that both terms have a common factor of [tex]\( 5 \times 2^n \)[/tex]:
[tex]\[ 10 \times 2^{n+2} = 10 \times 2^n \times 2^2 = 10 \times 4 \times 2^n = 40 \times 2^n. \][/tex]
[tex]\[ 5 \times 2^{n+3} = 5 \times 2^n \times 2^3 = 5 \times 8 \times 2^n = 40 \times 2^n. \][/tex]
So, the denominator becomes:
[tex]\[ 40 \times 2^n + 40 \times 2^n
= (40 + 40) \times 2^n
= 80 \times 2^n. \][/tex]
3. Divide the simplified numerator by the simplified denominator:
[tex]\[
\frac{240 \times 3^n}{80 \times 2^n} = \frac{240}{80} \times \frac{3^n}{2^n} = 3 \times \frac{3^n}{2^n}.
\][/tex]
Thus, we can express it as a single term:
[tex]\[
3 \times \left( \frac{3}{2} \right)^n = \frac{3^{n+1}}{2^n}.
\][/tex]
Therefore, the simplified form of the given expression is:
[tex]\[ \boxed{\frac{3^{n+1}}{2^n}}. \][/tex]
