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 exercises. Exercise 11: Given y = sqrt(25)-19 and x = sqrt(25)+19. Note that 2sqrt(5) ≈ 4.47. Since 2sqrt(5)-19 is negative, y is a complex number. We use i = sqrt(-1). So, y = sqrt(-(19-25)) = isqrt(19-25). And x = sqrt(19+25). a) Calculate xy Step 1: Substitute the expressions for x and y. xy = sqrt(19+25) · sqrt(25)-19 Step 2: Rewrite sqrt(25)-19 using i. sqrt(25)-19 = sqrt(-(19-25)) = isqrt(19-25) Step 3: Substitute and multiply. xy = sqrt(19+25) · isqrt(19-25) xy = i sqrt((19+25))(19-2sqrt(5)) Step 4: Apply the difference of squares formula (a+b)(a-b) = a^2-b^2. xy = i sqrt(19^2 - (25))^2 xy = i sqrt(361 - (4 · 5)) xy = i sqrt(361 - 20) xy = i sqrt(341) The value of xy is isqrt(341). b) Calculate (x-y)^2 Step 1: Expand (x-y)^2. (x-y)^2 = x^2 - 2xy + y^2 Step 2: Substitute x^2, y^2, and xy. x^2 = (sqrt(19+25))^2 = 19+2sqrt(5) y^2 = (sqrt(25)-19)^2 = 2sqrt(5)-19 xy = isqrt(341) (x-y)^2 = (19+2sqrt(5)) - 2(isqrt(341)) + (2sqrt(5)-19) Step 3: Simplify the expression. (x-y)^2 = 19+2sqrt(5) - 2isqrt(341) + 2sqrt(5)-19 (x-y)^2 = (19-19) + (2sqrt(5)+2sqrt(5)) - 2isqrt(341) (x-y)^2 = 4sqrt(5) - 2isqrt(341) The value of (x-y)^2 is 4sqrt(5) - 2isqrt(341). c) Calculate (x+y)^2 Step 1: Expand (x+y)^2. (x+y)^2 = x^2 + 2xy + y^2 Step 2: Substitute x^2, y^2, and xy. (x+y)^2 = (19+2sqrt(5)) + 2(isqrt(341)) + (2sqrt(5)-19) Step 3: Simplify the expression. (x+y)^2 = 19+2sqrt(5) + 2isqrt(341) + 2sqrt(5)-19 (x+y)^2 = (19-19) + (2sqrt(5)+2sqrt(5)) + 2isqrt(341) (x+y)^2 = 4sqrt(5) + 2isqrt(341) The value of (x+y)^2 is 4sqrt(5) + 2isqrt(341). d) Calculate (x+y)/(x-y) Step 1: Express (x+y)/(x-y) using the squared terms. (x+y)/(x-y) = sqrt(((x+y)^2)/((x-y)^2)) Step 2: Substitute the values of (x+y)^2 and (x-y)^2. (x+y)/(x-y) = sqrt(45) + 2isqrt(341)4sqrt(5) - 2isqrt(341) Step 3: Rationalize the denominator inside the square root by multiplying the numerator and denominator by the conjugate of the denominator, which is 4sqrt(5) + 2isqrt(341). (x+y)/(x-y) = sqrt((45) + 2isqrt(341))(4sqrt(5) + 2isqrt(341))(4sqrt(5) - 2isqrt(341))(4sqrt(5) + 2isqrt(341)) (x+y)/(x-y) = sqrt((45) + 2isqrt(341))^2(4sqrt(5))^2 - (2isqrt(341))^2 Calculate the denominator: (4sqrt(5))^2 - (2isqrt(341))^2 = (16 · 5) - (4 · i^2 · 341) = 80 - (4 · (-1) · 341) = 80 - (-1364) = 80 + 1364 = 1444. (x+y)/(x-y) = sqrt((45) + 2isqrt(341))^21444 Step 4: Simplify the square root. (x+y)/(x-y) = 4sqrt(5) + 2isqrt(341)sqrt(1444) Since sqrt(1444) = 38: (x+y)/(x-y) = 4sqrt(5) + 2isqrt(341)38 (x+y)/(x-y) = 2(2sqrt(5) + isqrt(341))38 (x+y)/(x-y) = 2sqrt(5) + isqrt(341)19 The value of (x+y)/(x-y) is 2sqrt(5) + isqrt(341)19. Exercise 12: a) Prove (a+b)^2 - (a-b)^2 = 4ab Step 1: Expand (a+b)^2 and (a-b)^2. (a+b)^2 = a^2+2ab+b^2 (a-b)^2 = a^2-2ab+b^2 Step 2: Substitute the expanded forms into the expression. (a+b)^2 - (a-b)^2 = (a^2+2ab+b^2) - (a^2-2ab+b^2) Step 3: Simplify the expression. = a^2+2ab+b^2 - a^2+2ab-b^2 = (a^2-a^2) + (b^2-b^2) + (2ab+2ab) = 0 + 0 + 4ab = 4ab Thus, (a+b)^2 - (a-b)^2 = 4ab is proven. b) Calculate 1000.001^2 - 999.999^2 Step 1: Use the identity (a+b)^2 - (a-b)^2 = 4ab. Let a+b = 1000.001 and a-b = 999.999. Step 2: Solve for a and b. Adding the two equations: (a+b) + (a-b) = 1000.001 + 999.999 2a = 2000 a = 1000 Subtracting the second equation from the first: (a+b) - (a-b) = 1000.001 - 999.999 2b = 0.002 b = 0.001 Step 3: Substitute the values of a and b into the identity. 1000.001^2 - 999.999^2 = 4 · a · b = 4 · (1000) · (0.001) Step 4: Calculate the result. = 4 · (1000 · (1)/(1000)) = 4 · 1 = 4 The value of 1000.001^2 - 999.999^2 is 4. Last free one today — make it count tomorrow, or type /upgrade for unlimited.