Question

solve. write your answer as an integer or a fraction in simplest form. 4 = 64^(3x + 4) x =

Explanation:

Step1: Express 64 as a power of 4

Since \( 64 = 4^3 \), we can rewrite the equation \( 4 = 64^{3x + 4} \) as \( 4 = (4^3)^{3x + 4} \).

Step2: Simplify the right - hand side using exponent rules

Using the exponent rule \( (a^m)^n=a^{m\times n} \), we have \( (4^3)^{3x + 4}=4^{3\times(3x + 4)} = 4^{9x+12} \). So the equation becomes \( 4^1=4^{9x + 12} \).

Step3: Set the exponents equal

If \( a^m=a^n \) (where \( a>0,a
eq1 \)), then \( m = n \). Since \( a = 4 \) here, we set \( 1=9x + 12 \).

Step4: Solve for x

Subtract 12 from both sides: \( 1-12=9x \), so \( - 11 = 9x \). Then divide both sides by 9: \( x=-\frac{11}{9} \).

Answer:

\(x = -\frac{11}{9}\)