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 for problems 6, 7, 8, 9, and 10. Problem 6: Write the given numbers in the form a+ib. a) 3i^3 - i^4 + 7i^3 - 10i^2 - 9 Step 1: Substitute powers of i (i^2=-1, i^3=-i, i^4=1). 3(-i) - (1) + 7(-i) - 10(-1) - 9 Step 2: Simplify and combine terms. -3i - 1 - 7i + 10 - 9 (-1+10-9) + (-3-7)i 0 - 10i The final answer is 0 - 10i. b) 2i^6 + ((2)/(-i))^3 + 5i^5 - 12i Step 1: Simplify powers of i (i^6=-1, i^5=i) and the fraction ((2)/(-i))^3. (2)/(-i) = (2i)/(-i^2) = (2i)/(1) = 2i (2i)^3 = 8i^3 = 8(-i) = -8i Step 2: Substitute and simplify. 2(-1) + (-8i) + 5(i) - 12i -2 - 8i + 5i - 12i Step 3: Combine real and imaginary parts. -2 + (-8+5-12)i -2 - 15i The final answer is -2 - 15i. Problem 7: Find the polar coordinate form of the complex numbers. a) z=i Step 1: Find modulus r and argument . x=0, y=1. r = sqrt(0^2+1^2) = 1 Since z=i is on the positive imaginary axis, = ()/(2). Step 2: Write in polar form r( + i). z = 1((()/(2)) + i(()/(2))) The final answer is (()/(2)) + i(()/(2)). b) z=3-4i Step 1: Find modulus r and argument . x=3, y=-4. r = sqrt(3^2+(-4)^2) = sqrt(9+16) = sqrt(25) = 5 The number is in the fourth quadrant. = ((-4)/(3)) = -((4)/(3)) Step 2: Write in polar form r( + i). z = 5((-((4)/(3))) + i(-((4)/(3)))) The final answer is 5((-((4)/(3))) + i(-((4)/(3)))). Problem 8: Find the values of x and y that satisfy the equation (x+y)+i(x-y)=14.8+6.2i. Step 1: Equate the real and imaginary parts. x+y = 14.8 (1) x-y = 6.2 (2) Step 2: Solve the system of equations. Add (1) and (2): (x+y)+(x-y) = 14.8+6.2 2x = 21 x = 10.5 Substitute x=10.5 into (1): 10.5+y = 14.8 y = 14.8-10.5 y = 4.3 The final answer is x=10.5, y=4.3. Problem 9: Express (-1+i) in the form re^i, where r is positive and - < < . Step 1: Find modulus r and argument . x=-1, y=1. r = sqrt((-1)^2+1^2) = sqrt(1+1) = sqrt(2) The number is in the second quadrant. The reference angle is ((1)/(1)) = ()/(4). = - ()/(4) = (3)/(4) This value is within (-, ). Step 2: Write in the form re^i. z = sqrt(2)e^i(3)/(4) The final answer is sqrt(2)e^i(3)/(4). Problem 10: If (a+b)+i(a-b)=(2+5i)^2+i(2-3i), find the values of a and b. Step 1: Simplify the right-hand side (RHS). (2+5i)^2 = 4 + 20i + 25i^2 = 4 + 20i - 25 = -21 + 20i i(2-3i) = 2i - 3i^2 = 2i - 3(-1) = 3 + 2i Add the simplified terms: (-21 + 20i) + (3 + 2i) = (-21+3) + (20+2)i = -18 + 22i Step 2: Equate the real and imaginary parts of the LHS and the simplified RHS. a+b = -18 (1) a-b = 22 (2) Step 3: Solve the system of equations. Add (1) and (2): (a+b)+(a-b) = -18+22 2a = 4 a = 2 Substitute a=2 into (1): 2+b = -18 b = -18-2 b = -20 The final answer is a=2, b=-20.