Question
Real Tutor Solution
Tutor-Verified Answer
Answer
\( x = 0.3 \)
Solution
Solve the equation \( 32^{2x}=8 \).
Solve the equation by following steps:
- step0: Solve for \(x\):
\(32^{2x}=8\)
- step1: Rewrite in exponential form:
\(2^{10x}=2^{3}\)
- step2: Set the exponents equal:
\(10x=3\)
- step3: Divide both sides:
\(\frac{10x}{10}=\frac{3}{10}\)
- step4: Divide the numbers:
\(x=\frac{3}{10}\)
The solution to the equation \(32^{2x}=8\) is \(x=\frac{3}{10}\) or \(x=0.3\).
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Simplify this solution Beyond the Answer
To solve the equation \( 32^{2x} = 8 \), we can express both sides in terms of powers of 2. We know that \( 32 = 2^5 \) and \( 8 = 2^3 \). Thus, we rewrite the equation as: \[ (2^5)^{2x} = 2^3 \] Using the power of a power property, we have: \[ 2^{10x} = 2^3 \] Since the bases are the same, we can equate the exponents: \[ 10x = 3 \] Now, solve for \( x \) by dividing both sides by 10: \[ x = \frac{3}{10} \] So, the solution is \( x = 0.3 \)!
