Certainly! Let's solve the equation step-by-step:
Given equation:
[tex]\[ 2^n \times 4 = \frac{1}{2^n} \times 8 \][/tex]
### Step 1: Simplify both sides of the equation
First, let's simplify the right side of the equation:
[tex]\[ \frac{1}{2^n} \times 8 = \frac{8}{2^n} \][/tex]
Now the equation looks like:
[tex]\[ 2^n \times 4 = \frac{8}{2^n} \][/tex]
### Step 2: Bring all terms involving [tex]\(2^n\)[/tex] to one side
Multiply both sides of the equation by [tex]\(2^n\)[/tex] to get rid of the fraction:
[tex]\[ (2^n \times 4) \times 2^n = \frac{8}{2^n} \times 2^n \][/tex]
This simplifies to:
[tex]\[ 4 \times (2^n \times 2^n) = 8 \][/tex]
### Step 3: Simplify the left side
Recall the property of exponents: [tex]\(2^n \times 2^n = 2^{2n}\)[/tex]. So our equation becomes:
[tex]\[ 4 \times 2^{2n} = 8 \][/tex]
### Step 4: Write 4 and 8 as powers of 2
We know that:
[tex]\[ 4 = 2^2 \][/tex]
[tex]\[ 8 = 2^3 \][/tex]
So the equation now is:
[tex]\[ 2^2 \times 2^{2n} = 2^3 \][/tex]
### Step 5: Simplify the expression
Combine the exponents on the left side:
[tex]\[ 2^{2 + 2n} = 2^3 \][/tex]
This gives us:
[tex]\[ 2^{2 + 2n} = 2^3 \][/tex]
### Step 6: Equate the exponents since the bases are the same
Since the bases are the same, the exponents must be equal:
[tex]\[ 2 + 2n = 3 \][/tex]
### Step 7: Solve for [tex]\(n\)[/tex]
Subtract 2 from both sides of the equation:
[tex]\[ 2n = 1 \][/tex]
Now divide by 2:
[tex]\[ n = \frac{1}{2} \][/tex]
So the solution to the equation is:
[tex]\[ n = \frac{1}{2} \][/tex]
