Try It! Use the Power of Power Property to Solve Equations With Rational Exponents 3. What is the solution of each equation? a. \( 256^{x+2}=4^{3 x+9} \)

To solve the equation \( 256^{x+2} = 4^{3x+9} \) using the power property with rational exponents, follow these steps: ### Step 1: Express Both Sides with the Same Base First, recognize that both 256 and 4 are powers of 2: - \( 256 = 2^8 \) - \( 4 = 2^2 \) Rewrite the equation using base 2: \[ (2^8)^{x+2} = (2^2)^{3x+9} \] ### Step 2: Apply the Power of a Power Property Use the exponentiation rule \((a^m)^n = a^{mn}\): \[ 2^{8(x+2)} = 2^{2(3x+9)} \] Simplify the exponents: \[ 2^{8x + 16} = 2^{6x + 18} \] ### Step 3: Set the Exponents Equal to Each Other Since the bases are the same, set the exponents equal: \[ 8x + 16 = 6x + 18 \] ### Step 4: Solve for \( x \) Subtract \( 6x \) from both sides: \[ 2x + 16 = 18 \] Subtract 16 from both sides: \[ 2x = 2 \] Divide both sides by 2: \[ x = 1 \] ### **Solution** The solution to the equation is **\( x = 1 \)**.