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We start with the equation:   (5/2)^x = (4/25)^2 Step 1. Rewrite (4/25) in terms of squares:   4 = 2^2 and 25 = 5^2, so   (4/25) = (2^2 / 5^2) = (2/5)^2. Step 2. Raise to the power of 2:   (4/25)^2 = [(2/5)^2]^2 = (2/5)^4. Step 3. Express (2/5)^4 in terms of (5/2):   Since (2/5) is the reciprocal of (5/2), we have:   (2/5) = (5/2)^(-1).   Thus, (2/5)^4 = [(5/2)^(-1)]^4 = (5/2)^(-4). Step 4. Substitute back into the equation:   (5/2)^x = (5/2)^(-4). Step 5. Since the bases are the same, the exponents must be equal:   x = -4. Thus, the solution is x = -4.

We start with the equation:   22x² + 2x − 0.5 = 42 Step 1. Rearrange the Equation Subtract 42 from both sides to set the equation equal to 0:   22x² + 2x − 0.5 − 42 = 0   22x² + 2x − 42.5 = 0 This is a quadratic equation of the form ax² + bx + c = 0 where:   a = 22, b = 2, c = −42.5 Step 2. Apply the Quadratic Formula The quadratic formula is:   x = [−b ± √(b² − 4ac)] / (2a) Plug in the values:   x = [−2 ± √(2² − 4·22·(−42.5))] / (2·22) Step 3. Evaluate the Discriminant Compute b² − 4ac:   b² = 4   4ac = 4 · 22 · (−42.5) = −3740 Therefore,   Discriminant = 4 − (−3740) = 4 + 3740 = 3744 Step 4. Simplify the Square Root We need to simplify √3744. Notice that:   3744 = 16

(5/2)^x=(4/25)^2 (5/2)^x=((5/2)^2)^-2 (5/2)^x=(5/2)^-4 x=-4

(9-8)^(-3)=1^(-3)=1/(1)^3