Question

quiz final exam practice
solve the equation.
$2^{x^2} = 128^x$
the solution set is
(type an exact answer, using radicals as needed. use a comma to separate answers as needed.)

Explanation:

Step1: Express 128 as a power of 2

We know that \(128 = 2^7\), so the equation \(2^{x^2}=128^x\) can be rewritten as \(2^{x^2}=(2^7)^x\).

Step2: Simplify the right - hand side

Using the power - of - a - power rule \((a^m)^n=a^{mn}\), we have \((2^7)^x = 2^{7x}\). So the equation becomes \(2^{x^2}=2^{7x}\).

Step3: Set the exponents equal

Since the bases are the same (and the base \(2>0,a
eq1\)), we can set the exponents equal to each other: \(x^2 = 7x\).

Step4: Solve the quadratic equation

Rearrange the equation \(x^2-7x = 0\). Factor out an \(x\): \(x(x - 7)=0\).
Using the zero - product property, if \(ab = 0\), then either \(a = 0\) or \(b = 0\). So \(x=0\) or \(x - 7=0\), which gives \(x = 0\) or \(x=7\).

Answer:

The solution set is \(\{0,7\}\)