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 questions. QUESTION 1 a) Evaluate: (i) (3^2 × 7^0)^3 Step 1: Evaluate the terms inside the parenthesis. Recall that any non-zero number raised to the power of 0 is 1. So, 7^0 = 1. 3^2 = 9 7^0 = 1 Step 2: Substitute these values back into the expression. (9 × 1)^3 9^3 Step 3: Calculate the final value. 9^3 = 9 × 9 × 9 = 81 × 9 = 729 The value is 729. (ii) (0.56)^0 Step 1: Apply the rule for exponents. Any non-zero number raised to the power of 0 is 1. (0.56)^0 = 1 The value is 1. b) Find the value of sqrt(81) Step 1: Find the number that, when multiplied by itself, equals 81. sqrt(81) = 9 The value is 9. c) Express 0.00698 in standard form. Step 1: Move the decimal point to the right until the number is between 1 and 10. The number becomes 6.98. Step 2: Count how many places the decimal point was moved. The decimal point was moved 3 places to the right. Step 3: Write the number in standard form (A × 10^n). Since the decimal was moved to the right, the exponent n is negative. 0.00698 = 6.98 × 10^-3 The standard form is 6.98 × 10^-3. QUESTION 2 a) Derive the quadratic formula by completing the square method, given an equation ax^2 + bx + c = 0, where a, b and c are constants. Step 1: Start with the general quadratic equation and divide by a (assuming a ≠ 0). ax^2 + bx + c = 0 x^2 + (b)/(a)x + (c)/(a) = 0 Step 2: Move the constant term to the right side of the equation. x^2 + (b)/(a)x = -(c)/(a) Step 3: Complete the square on the left side by adding ((b)/(2a))^2 to both sides. x^2 + (b)/(a)x + ((b)/(2a))^2 = -(c)/(a) + ((b)/(2a))^2 Step 4: Factor the left side as a perfect square and simplify the right side. (x + (b)/(2a))^2 = -(c)/(a) + (b^2)/(4a^2) To combine the terms on the right, find a common denominator: (x + (b)/(2a))^2 = -(4ac)/(4a^2) + (b^2)/(4a^2) (x + (b)/(2a))^2 = (b^2 - 4ac)/(4a^2) Step 5: Take the square root of both sides. x + (b)/(2a) = ±sqrt((b^2 - 4ac)/(4a^2)) x + (b)/(2a) = ±sqrt(b^2 - 4ac)sqrt(4a^2) x + (b)/(2a) = ±sqrt(b^2 - 4ac)2a Step 6: Isolate x to obtain the quadratic formula. x = -(b)/(2a) ± sqrt(b^2 - 4ac)2a x = -b ± sqrt(b^2 - 4ac)2a The quadratic formula is x = -b ± sqrt(b^2 - 4ac)2a. b) Solve for a and b in the equations below: (9)/(a) + (12)/(b) = -3 (1) (3)/(a) + (6)/(b) = 4 (2) Step 1: Let x = (1)/(a) and y = (1)/(b) to transform the system into linear equations. 9x + 12y = -3 (1') 3x + 6y = 4 (2') Step 2: Use the elimination method to solve for x and y. Multiply equation (2') by 3 to make the coefficients of x equal. 3 × (3x + 6y) = 3 × 4 9x + 18y = 12 (3') Step 3: Subtract equation (1') from equation (3'). (9x + 18y) - (9x + 12y) = 12 - (-3) 6y = 15 y = (15)/(6) y = (5)/(2) Step 4: Substitute the value of y into equation (2') to solve for x. 3x + 6((5)/(2)) = 4 3x + 3 × 5 = 4 3x + 15 = 4 3x = 4 - 15 3x = -11 x = -(11)/(3) Step 5: Convert back to a and b using a = (1)/(x) and b = (1)/(y). a = (1)/(-11)3 = -(3)/(11) b = (1)/(5)2 = (2)/(5) The values are a = -(3)/(11), b = (2)/(5). Drop the next question.