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10.6 Roots of Complex Numbers For complex numbers r and z and for any positive integer n, r is an nth root of z if and only if rn = z. Complex Roots Theorem For any positive integer n and any complex number z = rcisΞΈ, the n distinct nth roots of z are the complex numbers π βππππ π+2ππ π for k= 0, 1, 2, . . ., n-1 Example 1: 2π a) Find the four fourth roots of 81πππ 3 b) Find the three cube roots of 125πππ LESSON 10.6 ROOTS OF COMPLEX NUMBERS π 4 1 Example 2: a) Find the fifth roots of -32i b) Find the cube roots of 1000i c) Find the fifth roots of -3 + 3i LESSON 10.6 ROOTS OF COMPLEX NUMBERS 2 Example 3: a) Graph the four fourth roots of 81. imaginary real b) Graph the five fifth roots of 32. imaginary real LESSON 10.6 ROOTS OF COMPLEX NUMBERS 3 Example 4: a) Find the five fifth roots of 1 and locate them on the complex plane. imaginary real b) Find the three cube roots of 1 and locate them on the complex plane. imaginary real LESSON 10.6 ROOTS OF COMPLEX NUMBERS 4 Example 5: Use the cube roots of unity to find the cube roots of other real numbers. a) 125 b) 216 c) Find the fifth roots of 243 HOMEWORK: Page 525 Class Ex. #1-10 odd; 12 Page 526 Practice Ex. #1-10 even; 12-14; 16-20 LESSON 10.6 ROOTS OF COMPLEX NUMBERS 5 LESSON 10.6 ROOTS OF COMPLEX NUMBERS 6