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Roots of a complex number Chapter 7 review # 31 & 35 & 33 To view this power point, right click on the screen and choose Full screen.

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Find. Real number axis Imaginary number axis Graph the ordered pair (a,b). (-1,1) tan = 1 = arctan (-1) = -45 o = 315 o in quadrant IV or = 135 o in quadrant II. Change the number from a + bi form to r(cos + isin ) form. #31) Find the square root of -1 + i

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Change the number from a + bi form to r(cos + isin ) form. Since is in quadrant II, we will use 135 o. Now use the rule for finding the roots of a complex number. r (cos + isin ) n a + bi = n Where = o k. n and k = 0, 1, 2 … n – i = (cos 135 o + isin 135 o ) 2 r = (-1) 2 + (1) 2

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-1 + i = (cos 135 o + isin 135 o ) Since is in quadrant II, we will use 135 o. 2 (cos + isin ) -1 + i = = 135 o o k 2 4 or 2 (cos + isin 247 ) 4 = and k = 0 and 1 2 r = (-1) 2 + (1) 2 Change the number from a + bi form to r(cos + isin ) form. #31) Find the square root of -1 + i

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-1 + i = (cos + i sin ) Since is in quadrant II, we will use. 2 (cos + i sin ) -1 + i = = + 2 k 4 or 2 (cos + i sin ) 4 = and k = 0 and 1 2 r = (-1) 2 + (1) 2 The answer could also expressed in radians Change the number from a + bi form to r(cos + isin ) form. #31) Find the square root of -1 + i

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We use the same method to solve an equation. # 35) Find all complex solutions of the equation. x 4 – i = 0 x 4 = i Find the 4 th root of i. (Use 0 + i). r = (0) 2 + (1) 2 The ordered pair is (0,1) 0 + i = 1(cos 90 o + i sin 90 o )

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Find the 4 th root of i. 0 + i = (cos 90 o + isin 90 o ) cos + i sin 0 + i = = 90 o o k 4 4 or = cos + i sin 112 and k = 0, 1, 2, 3 = cos + i sin 202 = cos + i sin 292 or Change the number from a + bi form to r(cos + isin ) form.

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Find the 4 th root of the real number i = 81(cos 0 o + isin 0 o ) 3(cos 0 + i sin 0 81 = = 0 o o k 4 4 or = 3(cos 90 + i sin 90 ) = 3i and k = 0, 1, 2, 3 = 3(cos i sin 180 ) = –3 = 3(cos i sin 270 ) = –3i or Change the number from a + bi form to r(cos + isin ) form. #33)

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