Let's solve the equation [tex]\(2^n \times 4 = \frac{1}{2^n} \times 8\)[/tex] step by step.
1. Rewrite the equation:
[tex]\[
2^n \cdot 4 = \frac{1}{2^n} \cdot 8
\][/tex]
2. Simplify the right-hand side:
Since [tex]\(\frac{1}{2^n} \cdot 8\)[/tex] can be rewritten as:
[tex]\[
\frac{8}{2^n}
\][/tex]
So our equation becomes:
[tex]\[
2^n \cdot 4 = \frac{8}{2^n}
\][/tex]
3. Combine powers of 2 on both sides:
Multiply both sides by [tex]\(2^n\)[/tex] to clear the fraction:
[tex]\[
2^n \cdot 4 \cdot 2^n = 8
\][/tex]
This simplifies to:
[tex]\[
4 \cdot 2^{2n} = 8
\][/tex]
4. Isolate the term involving [tex]\(n\)[/tex]:
Divide both sides by 4:
[tex]\[
2^{2n} = \frac{8}{4}
\][/tex]
Simplify the right-hand side:
[tex]\[
2^{2n} = 2
\][/tex]
5. Express the right-hand side as a power of 2:
Notice that 2 can be expressed as [tex]\(2^1\)[/tex]:
[tex]\[
2^{2n} = 2^1
\][/tex]
6. Set the exponents equal to each other:
Since the bases are the same, we can set the exponents equal to each other:
[tex]\[
2n = 1
\][/tex]
7. Solve for [tex]\(n\)[/tex]:
Divide both sides by 2:
[tex]\[
n = \frac{1}{2}
\][/tex]
So, the solution to the equation [tex]\(2^n \times 4 = \frac{1}{2^n} \times 8\)[/tex] is:
[tex]\[
n = \frac{1}{2}
\][/tex]
