To simplify the expression [tex]\(\frac{16 \times 2^{n+1} - 4 \times 2^n}{16 \times 2^{n+2} - 2 \times 2^{n+2}}\)[/tex], we will follow these steps:
1. Simplify the numerator:
The numerator is given by:
[tex]\[ 16 \times 2^{n+1} - 4 \times 2^n \][/tex]
We can rewrite this by factoring out the common term [tex]\(2^n\)[/tex]:
[tex]\[ 16 \times 2^{n+1} - 4 \times 2^n = 16 \times 2 \times 2^n - 4 \times 2^n = 32 \times 2^n - 4 \times 2^n \][/tex]
Now, factor [tex]\(2^n\)[/tex] out of the expression:
[tex]\[ 32 \times 2^n - 4 \times 2^n = 2^n (32 - 4) = 2^n \times 28 = 28 \times 2^n \][/tex]
2. Simplify the denominator:
The denominator is given by:
[tex]\[ 16 \times 2^{n+2} - 2 \times 2^{n+2} \][/tex]
Similarly, factor out the common term [tex]\(2^{n+2}\)[/tex]:
[tex]\[ 16 \times 2^{n+2} - 2 \times 2^{n+2} = (16 - 2) \times 2^{n+2} = 14 \times 2^{n+2} = 14 \times 4 \times 2^n \][/tex]
Recognize that [tex]\(2^{n+2} = 4 \times 2^n\)[/tex], hence:
[tex]\[ 14 \times 4 \times 2^n = 56 \times 2^n \][/tex]
3. Simplify the entire fraction:
Now, our expression is:
[tex]\[ \frac{28 \times 2^n}{56 \times 2^n} \][/tex]
Here, [tex]\(2^n\)[/tex] in the numerator and denominator cancels out, leaving:
[tex]\[ \frac{28}{56} \][/tex]
Simplify [tex]\(\frac{28}{56}\)[/tex]:
[tex]\[ \frac{28}{56} = \frac{1}{2} \][/tex]
4. Final Answer:
The simplified form of the given expression is [tex]\(\boxed{\frac{1}{2}}\)[/tex].
