Practice with Multiplying and Dividing Complex Numbers Topic Index | Algebra2/Trig Index | Regents Exam Prep Center

Solve the following problems.  Answers are to be in simplest a+bi form.
You should be able to solve the problems both with, and without,

 1. Multiply:   (3 + 5i)(3 - 5i) Choose: 9 - 25i 25 34 Explanation Remember that a complex number times its conjugate will be a real number. (3+5i)(3-5i)=9-15i+15i-25i² =9-25(-1) = 9+25 = 34

 2. Multiply:  (8 + 9i)(7 - 3i) Choose: 15 - 12i 29 - 39i 83 + 39i Explanation (8+9i)(7-3i)=56-24i+63i-27i² =56+39i-27(-1) =83+39i

 3. Multiply:  (4 - 3i)(3 - 4i) Choose: 25 -25i 12 - 12i Explanation (4-3i)(3-4i)=12-16i-9i+12i² =12-25i+12(-1) =-25i

 4. Simplify:  (2 + 5i)2 Choose: 21 + 20i -21 + 20i 29 + 20i Explanation (2+5i)²=(2+5i)(2+5i) =4+10i+10i+25i² =4+20i+25(-1) =4+20i-25=-21+20i

 5. Simplify:  8 + i(8 - i) Choose: 7 + 8i 8 + 8i 9 + 8i Explanation Be careful. Only the i will be distributed over the parentheses. 8+i(8-i)=8+8i-i² =8+8i-(-1)=9+8i

 6. Simplify: Choose: 5 - 2i 3 + 2i 15 + 10i Explanation Multiply the top and bottom by the conjugate 1+2i. The numerator becomes (7-4i)(1+2i)=7+14i-4i-8i² =7+10i-8(-1)=15+10i. The denominator becomes (1-2i)(1+2i)=1+2i-2i-4i² =1-4(-1)=1+4=5 Answer: (15+10i)/5=3+2i

 7. Simplify: Choose: 35/37 + (12/37)i 35 + 12i 35/36 + (12/36)i Explanation Multiply top and bottom by 6+i. The numerator will be (6+i)(6+i) =36+6i+6i+i² =36+12i+(-1)=35+12i The denominator will be (6-i)(6+i) =36+6i-6i-i²=36-(-1)=37 The answer will be 35/37 + 12/37 i.

 8. Simplify: Choose: 5 + 3i -5 - 3i 5 - 3i Explanation Multiply top and bottom by -i. The numerator will be -3i+5i²=-3i+5(-1)=-3i-5=-5-3i The denominator will be i(-i)=-i²=-(-1)=1 Answer will be -3i-5

 9. Simplify: Choose: 2/15 + i/15 2/15 - i/15 1/45 + i/15 Explanation Multiply top and bottom by 6+3i. The numerator will be 6+3i. The denominator will be (6-3i)(6+3i)=36-18i+18i-9i² =36-9(-1)=45 The answer is 6/45 + 3i/45 = 2/15 + i/15

 10. What is the multiplicative inverse of Choose: 2/(1 + i) (1/2) - (1/2)i (1 + i)/2 Explanation The multiplicative identity element for complex numbers is 1+0i. So (1/2 + 1/2 i)•(what???)= 1 + 0i The multiplicative inverse of a complex number is one over the number. Since our number can be written (1+i)/2 the inverse will be 2/(1+i)

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