This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Here are the solutions to the problems: 1) Simplify a) e^ x^2 Step 1: Use the property e^ A = A. In this case, A = x^2. e^ x^2 = x^2 The simplified expression is x^2. b) 4 + 25 Step 1: Use the logarithm property A + B = (A × B). 4 + 25 = (4 × 25) Step 2: Perform the multiplication. (4 × 25) = 100 Step 3: Evaluate 100. (Assuming base 10 for when no base is specified). _10 100 = 2 The simplified expression is 2. 2) Solve for the equation a) 3 _2 x + _2 x = 4 Step 1: Combine the logarithmic terms. 4 _2 x = 4 Step 2: Divide both sides by 4. _2 x = 1 Step 3: Convert the logarithmic equation to an exponential equation using the definition _b A = C b^C = A. x = 2^1 x = 2 The solution is x=2. b) 2e^-x + e^x = 3 Step 1: Rewrite e^-x as (1)/(e^x). (2)/(e^x) + e^x = 3 Step 2: Let y = e^x. Substitute y into the equation. (2)/(y) + y = 3 Step 3: Multiply the entire equation by y to eliminate the denominator. 2 + y^2 = 3y Step 4: Rearrange the terms to form a standard quadratic equation. y^2 - 3y + 2 = 0 Step 5: Factor the quadratic equation. (y-1)(y-2) = 0 Step 6: Solve for y. y-1 = 0 y = 1 y-2 = 0 y = 2 Step 7: Substitute back e^x = y and solve for x. For y=1: e^x = 1 x = 1 x = 0 For y=2: e^x = 2 x = 2 The solutions are x=0, 2. 3) Given that (x-2), (x+1), and (7x-2) are the first 3 consecutive terms of an AP. a) Find the value of x and the common difference. Step 1: In an arithmetic progression (AP), the common difference between consecutive terms is constant. Therefore, we can equate the differences between the terms. (x+1) - (x-2) = (7x-2) - (x+1) Step 2: Simplify both sides of the equation. x+1-x+2 = 7x-2-x-1 3 = 6x-3 Step 3: Solve for x. 3+3 = 6x 6 = 6x x = 1 Step 4: Find the common difference (d) by substituting x=1 into the expression for the difference between terms. d = (x+1) - (x-2) d = (1+1) - (1-2) d = 2 - (-1) d = 3 The value of x is 1 and the common difference is 3. b) Find the range of values of x. In this problem, x is a specific value that makes the given terms an arithmetic progression. There is no range of values for x. The value of x is the single value found in part (a). The value of x is 1.