Question

4^{2x + 11} = 1024

Explanation:

Step1: Express bases as powers of 2

Since \(4 = 2^2\) and \(1024 = 2^{10}\), rewrite the equation:
\((2^2)^{2x + 11}=2^{10}\)

Step2: Simplify left - hand side exponent

Using the power - of - a - power rule \((a^m)^n=a^{mn}\), we have:
\(2^{2(2x + 11)}=2^{10}\)
Simplify the exponent \(2(2x + 11)\):
\(2^{4x+22}=2^{10}\)

Step3: Set exponents equal

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

Step4: Solve for x

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

Answer:

\(x = - 3\)