Of course! Let's solve the equation step-by-step:
Given:
[tex]\[ 2^{x+2} = \frac{1}{32} \][/tex]
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### Step-by-Step Solution:
1. Understanding the equation:
The equation is given in exponential form with a base of 2. We know that [tex]\( \frac{1}{32} \)[/tex] can be expressed as a power of 2.
2. Express [tex]\(\frac{1}{32}\)[/tex] as a power of 2:
We know that:
[tex]\[ 32 = 2^5 \][/tex]
Therefore:
[tex]\[ \frac{1}{32} = \frac{1}{2^5} = 2^{-5} \][/tex]
3. Rewrite the equation with common bases:
Replace [tex]\(\frac{1}{32}\)[/tex] with [tex]\(2^{-5}\)[/tex] in the original equation:
[tex]\[ 2^{x+2} = 2^{-5} \][/tex]
4. Set the exponents equal to each other:
Since the bases are equal, we can set the exponents equal to each other:
[tex]\[ x + 2 = -5 \][/tex]
5. Solve for [tex]\(x\)[/tex]:
Isolate [tex]\(x\)[/tex] by subtracting 2 from both sides of the equation:
[tex]\[ x + 2 - 2 = -5 - 2 \][/tex]
So,
[tex]\[ x = -7 \][/tex]
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### Conclusion:
The solution to the equation [tex]\(2^{x+2} = \frac{1}{32}\)[/tex] is:
[tex]\[ x = -7 \][/tex]
By following this step-by-step process, we have found that the value of [tex]\(x\)[/tex] that satisfies the equation is [tex]\(-7\)[/tex].
