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Sagot :
Certainly! Let's simplify the given expression step by step:
[tex]\[ \frac{16 \times 2^{n+1} - 4 \times 2^n}{16 \times 2^{n+2} - 2 \times 2^{n+2}} \][/tex]
### Step 1: Simplify the Numerator
First, we simplify the numerator, which is [tex]\( 16 \times 2^{n+1} - 4 \times 2^n \)[/tex].
1. Recall the property of exponents: [tex]\( a \times 2^b = 2^{b+\log_2a} \)[/tex].
2. Rewrite [tex]\( 16 \times 2^{n+1}\)[/tex]:
[tex]\[ 16 = 2^4 \implies 16 \times 2^{n+1} = 2^4 \times 2^{n+1} = 2^{4+n+1} = 2^{n+5} \][/tex]
3. Rewrite [tex]\( 4 \times 2^n \)[/tex]:
[tex]\[ 4 = 2^2 \implies 4 \times 2^n = 2^2 \times 2^n = 2^{n+2} \][/tex]
4. Substitute these back into the numerator:
[tex]\[ 2^{n+5} - 2^{n+2} \][/tex]
### Step 2: Factor the Numerator
Factor out the common term [tex]\( 2^{n+2} \)[/tex]:
[tex]\[ 2^{n+2}(2^3 - 1) \quad \text{(since \( 2^{n+5} = 2^{n+2+3} = 2^{n+2} \times 2^3\))} \][/tex]
[tex]\[ 2^{n+2}(8 - 1) = 2^{n+2} \times 7 \][/tex]
So, the simplified numerator is [tex]\( 7 \times 2^{n+2} \)[/tex].
### Step 3: Simplify the Denominator
Now, simplify the denominator, which is [tex]\( 16 \times 2^{n+2} - 2 \times 2^{n+2} \)[/tex].
1. Rewrite [tex]\( 16 \times 2^{n+2} \)[/tex]:
[tex]\[ 16 = 2^4 \implies 16 \times 2^{n+2} = 2^4 \times 2^{n+2} = 2^{n+4+2} = 2^{n+6} \][/tex]
2. Rewrite [tex]\( 2 \times 2^{n+2} \)[/tex]:
[tex]\[ 2 \times 2^{n+2} = 2^1 \times 2^{n+2} = 2^{1+n+2} = 2^{n+3} \][/tex]
3. Substitute these into the denominator:
[tex]\[ 2^{n+6} - 2^{n+3} \][/tex]
### Step 4: Factor the Denominator
Factor out the common term [tex]\( 2^{n+3} \)[/tex]:
[tex]\[ 2^{n+3}(2^3 - 1) \quad \text{(since \( 2^{n+6} = 2^{3+n+3} = 2^{n+3} \times 2^3 \))} \][/tex]
[tex]\[ 2^{n+3}(8 - 1) = 2^{n+3} \times 7 \][/tex]
So, the simplified denominator is [tex]\( 7 \times 2^{n+3} \)[/tex].
### Step 5: Simplify the Whole Fraction
Now, let's combine the simplified numerator and denominator:
[tex]\[ \frac{7 \times 2^{n+2}}{7 \times 2^{n+3}} \][/tex]
Cancel out the common factors [tex]\( 7 \times 2^{n+2} \)[/tex] in the numerator and denominator:
[tex]\[ \frac{1}{2} = 0.5 \][/tex]
Thus, the original expression simplifies to:
[tex]\[ \frac{16 \times 2^{n+1} - 4 \times 2^n}{16 \times 2^{n+2} - 2 \times 2^{n+2}} = 0.5 \][/tex]
The values for the numerator and denominator are [tex]\( 28 \)[/tex] and [tex]\( 56 \)[/tex], respectively, giving results:
[tex]\[ (28, 56, 0.5) \][/tex]
[tex]\[ \frac{16 \times 2^{n+1} - 4 \times 2^n}{16 \times 2^{n+2} - 2 \times 2^{n+2}} \][/tex]
### Step 1: Simplify the Numerator
First, we simplify the numerator, which is [tex]\( 16 \times 2^{n+1} - 4 \times 2^n \)[/tex].
1. Recall the property of exponents: [tex]\( a \times 2^b = 2^{b+\log_2a} \)[/tex].
2. Rewrite [tex]\( 16 \times 2^{n+1}\)[/tex]:
[tex]\[ 16 = 2^4 \implies 16 \times 2^{n+1} = 2^4 \times 2^{n+1} = 2^{4+n+1} = 2^{n+5} \][/tex]
3. Rewrite [tex]\( 4 \times 2^n \)[/tex]:
[tex]\[ 4 = 2^2 \implies 4 \times 2^n = 2^2 \times 2^n = 2^{n+2} \][/tex]
4. Substitute these back into the numerator:
[tex]\[ 2^{n+5} - 2^{n+2} \][/tex]
### Step 2: Factor the Numerator
Factor out the common term [tex]\( 2^{n+2} \)[/tex]:
[tex]\[ 2^{n+2}(2^3 - 1) \quad \text{(since \( 2^{n+5} = 2^{n+2+3} = 2^{n+2} \times 2^3\))} \][/tex]
[tex]\[ 2^{n+2}(8 - 1) = 2^{n+2} \times 7 \][/tex]
So, the simplified numerator is [tex]\( 7 \times 2^{n+2} \)[/tex].
### Step 3: Simplify the Denominator
Now, simplify the denominator, which is [tex]\( 16 \times 2^{n+2} - 2 \times 2^{n+2} \)[/tex].
1. Rewrite [tex]\( 16 \times 2^{n+2} \)[/tex]:
[tex]\[ 16 = 2^4 \implies 16 \times 2^{n+2} = 2^4 \times 2^{n+2} = 2^{n+4+2} = 2^{n+6} \][/tex]
2. Rewrite [tex]\( 2 \times 2^{n+2} \)[/tex]:
[tex]\[ 2 \times 2^{n+2} = 2^1 \times 2^{n+2} = 2^{1+n+2} = 2^{n+3} \][/tex]
3. Substitute these into the denominator:
[tex]\[ 2^{n+6} - 2^{n+3} \][/tex]
### Step 4: Factor the Denominator
Factor out the common term [tex]\( 2^{n+3} \)[/tex]:
[tex]\[ 2^{n+3}(2^3 - 1) \quad \text{(since \( 2^{n+6} = 2^{3+n+3} = 2^{n+3} \times 2^3 \))} \][/tex]
[tex]\[ 2^{n+3}(8 - 1) = 2^{n+3} \times 7 \][/tex]
So, the simplified denominator is [tex]\( 7 \times 2^{n+3} \)[/tex].
### Step 5: Simplify the Whole Fraction
Now, let's combine the simplified numerator and denominator:
[tex]\[ \frac{7 \times 2^{n+2}}{7 \times 2^{n+3}} \][/tex]
Cancel out the common factors [tex]\( 7 \times 2^{n+2} \)[/tex] in the numerator and denominator:
[tex]\[ \frac{1}{2} = 0.5 \][/tex]
Thus, the original expression simplifies to:
[tex]\[ \frac{16 \times 2^{n+1} - 4 \times 2^n}{16 \times 2^{n+2} - 2 \times 2^{n+2}} = 0.5 \][/tex]
The values for the numerator and denominator are [tex]\( 28 \)[/tex] and [tex]\( 56 \)[/tex], respectively, giving results:
[tex]\[ (28, 56, 0.5) \][/tex]
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