Rewrite the bases 9 and 27 as powers of 3, and express the square root on the right side as a fractional exponent.

To find the value of x, we need to solve the given equation: 9^sqrt(x) = sqrt(27^x) Step 1: Rewrite the bases 9 and 27 as powers of 3, and express the square root on the right side as a fractional exponent. We know that 9 = 3^2 and 27 = 3^3. Also, sqrt(27^x) = (27^x)^1/2. Substitute these into the equation: (3^2)^sqrt(x) = ((3^3)^x)^1/2 Step 2: Apply the exponent rule (a^b)^c = a^bc to simplify both sides. 3^2sqrt(x) = (3^3x)^1/2 3^2sqrt(x) = 3^(3x)/(2) Step 3: Since the bases are equal, equate the exponents. 2sqrt(x) = (3x)/(2) Step 4: Solve the equation for x. Multiply both sides by 2: 4sqrt(x) = 3x Square both sides to eliminate the square root: (4sqrt(x))^2 = (3x)^2 16x = 9x^2 Rearrange the equation to form a quadratic equation: 9x^2 - 16x = 0 Factor out x: x(9x - 16) = 0 This gives two possible solutions: x = 0 or 9x - 16 = 0 If 9x - 16 = 0: 9x = 16 x = (16)/(9) Step 5: Check both solutions in the original equation. For x=0: 9^sqrt(0) = sqrt(27^0) 9^0 = sqrt(1) 1 = 1 So, x=0 is a valid solution. For x=(16)/(9): Left side: 9^sqrt((16)/(9)) = 9^(4)/(3) = (3^2)^(4)/(3) = 3^(8)/(3) Right side: sqrt(27^(16)/(9)) = (27^(16)/(9))^1/2 = ((3^3)^(16)/(9))^1/2 = (3^3 · (16)/(9))^1/2 = (3^(16)/(3))^1/2 = 3^(16)/(3) · (1)/(2) = 3^(8)/(3) Since 3^(8)/(3) = 3^(8)/(3), x=(16)/(9) is also a valid solution. The values of x are: x = 0, (16)/(9) 3 done, 2 left today. You're making progress.