Question
) \( 2^{\frac{x}{3}}=\frac{1}{128} \) Solve for \( x \)
Ask by Burns Schwartz. in South Africa
Feb 21,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( x = -21 \)
Solution
Solve the equation \( 2^{\frac{x}{3}}=\frac{1}{128} \).
Solve the equation by following steps:
- step0: Solve for \(x\):
\(2^{\frac{x}{3}}=\frac{1}{128}\)
- step1: Rewrite in exponential form:
\(2^{\frac{x}{3}}=2^{-7}\)
- step2: Set the exponents equal:
\(\frac{x}{3}=-7\)
- step3: Cross multiply:
\(x=3\left(-7\right)\)
- step4: Simplify the equation:
\(x=-21\)
The solution to the equation \(2^{\frac{x}{3}}=\frac{1}{128}\) is \(x=-21\).
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Simplify this solution Beyond the Answer
To solve the equation \( 2^{\frac{x}{3}}=\frac{1}{128} \), we start by recognizing that \( 128 \) can be expressed as a power of \( 2 \): \( 128 = 2^7 \). Therefore, we can rewrite \( \frac{1}{128} \) as \( 2^{-7} \). This transforms the equation into: \[ 2^{\frac{x}{3}} = 2^{-7} \] Since the bases are the same, we can set the exponents equal to each other: \[ \frac{x}{3} = -7 \] Now, solve for \( x \) by multiplying both sides by \( 3 \): \[ x = -21 \] Thus, the solution is \( x = -21 \).
