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6. Write the given numbers in the form a+ib. a) 3i^5 - i^4 + 7i^3 - 10i^2 - 9 Recall the powers of i: i^1=i, i^2=-1, i^3=-i, i^4=1. For i^5, we have i^5 = i^4 · i = 1 · i = i. Substitute these values into the expression: 3(i) - (1) + 7(-i) - 10(-1) - 9 = 3i - 1 - 7i + 10 - 9 Group the real and imaginary terms: = (-1 + 10 - 9) + (3i - 7i) = (0) + (-4i) = -4i The expression in the form a+ib is 0 - 4i. b) 2i^6 + ((2)/(-i))^3 + 5i^5 - 12i First, evaluate the powers of i: i^6 = i^4 · i^2 = 1 · (-1) = -1 i^5 = i^4 · i = 1 · i = i Next, simplify the term ((2)/(-i))^3: (2)/(-i) = (2)/(-i) · (i)/(i) = (2i)/(-i^2) = (2i)/(-(-1)) = (2i)/(1) = 2i Now, cube this result: (2i)^3 = 2^3 · i^3 = 8 · (-i) = -8i Substitute all simplified terms back into the original expression: 2(-1) + (-8i) + 5(i) - 12i = -2 - 8i + 5i - 12i Group the real and imaginary terms: = (-2) + (-8i + 5i - 12i) = -2 + (-15i) = -2 - 15i The expression in the form a+ib is -2 - 15i. 7. Find the polar coordinate form of the complex numbers. The polar form of a complex number z = a+bi is z = r( + i), where r = |z| = sqrt(a^2+b^2) and = (z). a) z = i This can be written as z = 0 + 1i. Calculate the modulus r: r = sqrt(0^2 + 1^2) = sqrt(1) = 1 Calculate the argument : The complex number i lies on the positive imaginary axis. = ()/(2) So, the polar coordinate form is 1((()/(2)) + i(()/(2))). b) z = 3 - 4i This is in the form a+bi with a=3 and b=-4. Calculate the modulus r: r = sqrt(3^2 + (-4)^2) = sqrt(9 + 16) = sqrt(25) = 5 Calculate the argument : The complex number 3-4i is in the fourth quadrant. = ((-4)/(3)) Using a calculator, ≈ -0.927 radians (or approximately -53.13^). So, the polar coordinate form is 5((-0.927) + i(-0.927)). 8. Find the values of x and y that satisfy the equation (x+y) + i(x-y) = 14.8 + 6.2i. For two complex numbers to be equal, their real parts must be equal and their imaginary parts must be equal. Step 1: Equate the real parts. x+y = 14.8 (1) Step 2: Equate the imaginary parts. x-y = 6.2 (2) Step 3: Solve the system of linear equations. Add equation (1) and equation (2): (x+y) + (x-y) = 14.8 + 6.2 2x = 21 x = (21)/(2) x = 10.5 Substitute the value of x into equation (1): 10.5 + y = 14.8 y = 14.8 - 10.5 y = 4.3 The values are x=10.5, y=4.3. 9. Express (-1+i) in the form re^i, where r is positive and - < < . Let z = -1+i. Step 1: Find the modulus r. r = |-1+i| = sqrt((-1)^2 + (1)^2) = sqrt(1+1) = sqrt(2) Step 2: Find the argument . The complex number -1+i corresponds to the point (-1,1) in the complex plane, which is in the second quadrant. The reference angle is given by = |(1)/(-1)| = 1. So, = ()/(4). Since the point is in the second quadrant, the argument is: = - = - ()/(4) = (3)/(4) This value of = (3)/(4) satisfies the condition - < < . Step 3: Write in the form re^i. z = sqrt(2)e^i(3)/(4) The expression is sqrt(2)e^i(3)/(4).