Question

4^{2x + 11} = 1024

Explanation:

Step1: Express both sides as powers of 2

Since \(4 = 2^2\) and \(1024 = 2^{10}\), we can rewrite the equation:
\((2^2)^{2x + 11}=2^{10}\)
Using the exponent rule \((a^m)^n=a^{mn}\), we get:
\(2^{2(2x + 11)} = 2^{10}\)
Simplify the exponent on the left:
\(2^{4x + 22}=2^{10}\)

Step2: Set the exponents equal

Since the bases are the same and the equation holds, the exponents must be equal:
\(4x + 22 = 10\)

Step3: Solve for x

Subtract 22 from both sides:
\(4x = 10 - 22\)
\(4x = -12\)
Divide both sides by 4:
\(x=\frac{-12}{4}\)
\(x = - 3\)

Answer:

\(x=-3\)