Question

solve for x.
64^{x + 5} = 8^{4x}

x = \boxed{}

Explanation:

Step1: Express bases as powers of 2

We know that \(64 = 2^6\) and \(8 = 2^3\). So we can rewrite the equation \(64^{x + 5}=8^{4x}\) as \((2^6)^{x + 5}=(2^3)^{4x}\).

Step2: Apply exponent power rule

Using the power - of - a - power rule \((a^m)^n=a^{mn}\), we get \(2^{6(x + 5)}=2^{3\times4x}\), which simplifies to \(2^{6x+30}=2^{12x}\).

Step3: Set exponents equal

Since the bases are the same (\(a^m=a^n\) implies \(m = n\) for \(a>0,a
eq1\)), we can set the exponents equal to each other: \(6x + 30=12x\).

Step4: Solve for x

Subtract \(6x\) from both sides: \(30 = 12x-6x\), which gives \(6x=30\). Then divide both sides by 6: \(x=\frac{30}{6}=5\).

Answer:

\(x = 5\)