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Chapter 4: Problem 9

Solve each exponential equation by expressing each side as a power of the same base and then equating exponents. $$32^{x}=8$$

Short Answer

Expert verified
The solution to the equation \(32^x = 8\) is \(x = \frac{3}{5}\)

Step by step solution

01

Express in Power of the Same Base

Rewrite both 32 and 8 as powers of 2. So, \(32^x = 8\) transforms to \((2^5)^x = 2^3\).
02

Simplify

By knowing the rule of exponent which states that \((a^m)^n = a^{mn}\) we simplify to get \(2^{5x} = 2^3\).
03

Equating Exponents

Having the same bases (2's) on both sides of the equation, we can set the exponents equal to each other. This gives \(5x = 3\).
04

Solving for variable

Finally, solve the equation for \(x\). We do this by dividing each side by 5 to get \(x = \frac{3}{5}\).

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