Westonci.ca is your go-to source for answers, with a community ready to provide accurate and timely information. Get quick and reliable solutions to your questions from knowledgeable professionals on our comprehensive Q&A platform. Connect with a community of professionals ready to provide precise solutions to your questions quickly and accurately.
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]
We hope this was helpful. Please come back whenever you need more information or answers to your queries. Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. Thank you for using Westonci.ca. Come back for more in-depth answers to all your queries.
