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Solve for [tex]\( n \)[/tex]:

[tex]\[ 2^n \times 4 = \frac{1}{2^n} \times 8 \][/tex]

Sagot :

GinnyAnsweridn
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]
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