Question

(\frac{1}{512} = 2^{3x - 1})

Explanation:

Step1: Express 512 as a power of 2

We know that \(512 = 2^9\), so \(\frac{1}{512}=\frac{1}{2^9}\). By the negative exponent rule \(a^{-n}=\frac{1}{a^n}\), we can rewrite \(\frac{1}{2^9}\) as \(2^{-9}\). So the equation becomes \(2^{-9}=2^{3x - 1}\).

Step2: Set the exponents equal

Since the bases are the same (both are 2) and the equation holds, we can set the exponents equal to each other (for \(a^m=a^n\), where \(a>0,a
eq1\), then \(m = n\)). So we have the equation \(-9=3x - 1\).

Step3: Solve for x

First, add 1 to both sides of the equation: \(-9 + 1=3x-1 + 1\), which simplifies to \(-8 = 3x\). Then divide both sides by 3: \(x=\frac{-8}{3}\).

Answer:

\(x = -\frac{8}{3}\)