Question

solve the equation $8^{3x} = 4^{x+1}$. if no solution exists, enter none.
x =

Explanation:

Step1: Express bases as powers of 2

Since \(8 = 2^3\) and \(4 = 2^2\), rewrite the equation:
\((2^3)^{3x} = (2^2)^{x + 1}\)

Step2: Simplify exponents

Using the power - of - a - power rule \((a^m)^n=a^{mn}\), we get:
\(2^{9x}=2^{2(x + 1)}\)
Simplify the right - hand side exponent: \(2(x + 1)=2x+2\), so the equation becomes \(2^{9x}=2^{2x + 2}\)

Step3: Set exponents equal

If \(a^m=a^n\) (where \(a>0,a
eq1\)), then \(m = n\). Since \(a = 2>0,a
eq1\), we set the exponents equal:
\(9x=2x + 2\)

Step4: Solve for x

Subtract \(2x\) from both sides:
\(9x-2x=2x + 2-2x\)
\(7x=2\)
Divide both sides by 7:
\(x=\frac{2}{7}\)

Answer:

\(x = \frac{2}{7}\)