Question

solve the equation $2^{x + 1}2^{x} = 64$. if no solution exists, enter none.
x =

Explanation:

Step1: Use exponent rule \(a^m \cdot a^n = a^{m + n}\)

For \(2^{x + 1} \cdot 2^{x}\), apply the rule: \(2^{(x + 1)+x}=2^{2x + 1}\)
So the equation becomes \(2^{2x + 1}=64\)

Step2: Rewrite 64 as a power of 2

Since \(64 = 2^6\), the equation is \(2^{2x + 1}=2^6\)

Step3: Set exponents equal (if bases are equal)

If \(a^m=a^n\) and \(a>0,a
eq1\), then \(m = n\). So \(2x + 1=6\)

Step4: Solve for x

Subtract 1 from both sides: \(2x=6 - 1=5\)
Divide by 2: \(x=\frac{5}{2}=2.5\)

Answer:

\(x = \frac{5}{2}\) (or \(2.5\))