129 1.3 Complex Numbers EXAMPLE 7 Simplifying Powers of i Simplify each power of i. (a) i15 (b) i -3 SOLUTION (a) Because i4 = 1, write the given power as a product involving i4. i15 = i12 # i3 = 1i423 # i3 = 131-i2 = -i (b) Multiply i-3 by 1 in the form of i4 to create the least positive exponent for i. i-3 = i-3 # 1 = i-3 # i4 = i i4 = 1 S Now Try Exercises 89 and 97. 1.3 Exercises CONCEPT PREVIEW Fill in the blank(s) to correctly complete each sentence. 1. By definition, i = , and therefore, i2 = . 2. If a and b are real numbers, then any number of the form a + bi is a(n) . 3. The numbers 6 + 5i and 6 - 5i, which differ only in the sign of their imaginary parts, are . 4. The product of a complex number and its conjugate is always a(n) . 5. To find the quotient of two complex numbers in standard form, multiply both the numerator and the denominator by the complex conjugate of the . Concept Check Identify each number as real, complex, pure imaginary, or nonreal complex. (More than one of these descriptions will apply.) 11. -4 12. 0 13. 13i 14. -7i 15. 5 + i 16. -6 - 2i 17. p 18. 224 19. 2-25 20. 2-36 Write each number as the product of a real number and i. See Example 1. 21. 2-25 22. 2-36 23. 2-10 24. 2-15 25. 2-288 26. 2-500 27. -2-18 28. -2-80 Find each product or quotient. Simplify the answers. See Example 2. 29. 2-13 # 2-13 30. 2-17 # 2-17 31. 2-3 # 2-8 32. 2-5 # 2-15 33. 2-30 2-10 34. 2-70 2-7 CONCEPT PREVIEW Decide whether each statement is true or false. If false, correct the right side of the equation. 6. 2-25 = 5i 7. 2-4 # 2-9 = -6 8. i12 = 1 9. 1-2 + 7i2 - 110 - 6i2 = -12 + i 10. 15 + 3i22 = 16
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