Type the correct answer in the box. Use numerals instead of words. If necessary, use \( / \) for the fraction bar. What value of \( x \) satisfies this equation? \( 1.5(4)^{2 x}=12 \) Round your answer to the nearest hundredth. The value of \( x \) is

To solve the equation \( 1.5(4)^{2 x}=12 \), we first divide both sides by 1.5: \[ (4)^{2 x} = \frac{12}{1.5} = 8 \] Next, we recognize that \( 8 \) can be expressed as \( 4^{3/2} \) (because \( 4^{3/2} = (2^2)^{3/2} = 2^3 = 8 \)). Thus, we can rewrite the equation as: \[ (4)^{2 x} = (4)^{3/2} \] Since the bases are the same, we can equate the exponents: \[ 2 x = \frac{3}{2} \] Now, divide both sides by 2 to solve for \( x \): \[ x = \frac{3/2}{2} = \frac{3}{4} \] To provide the answer rounded to the nearest hundredth, we get: \[ x = 0.75 \] So, the value of \( x \) is \( 0.75 \).