4 Square Roots Finding Square Roots 3 2 = 9 (-3) 2 = 9N.B. -3 2 = -9 (½) 2 = (¼) The square root of 9 is 3 The square root of 9 is also –3 The square root of (¼) is (½)
5 Square Roots The square root symbol Radical sign The expression within is the radicand Square Root If a is a positive number, then is the positive square root of a is the negative square root of a Also,
6 Approximating Square Roots Perfect squares are numbers whose square roots are integers, for example 81 = 9 2. Square roots of other numbers are irrational numbers, for example We can approximate square roots with a calculator.
7 Approximating Square Roots 3.162(Calculator) We can determine that it is greater than 3 and less then 4 because 3 2 = 9 and 4 2 =16.
8 Cube Roots 2 is the cube root of 8 because 2 3 = 8. 8 and 2 3 above are radicands 3 is called the index (index 2 is omitted).
9 Cube Roots Evaluated 2 is the cube root of 8 because 2 3 = 8. 8 and 2 3 above are radicands 3 is called the index (index 2 is omitted)
10 nth Roots The number b is an nth root of a,, if b n = a.
11 nth Roots An nth root of number a is a number whose nth power is a. a number whose nth power is a If the index n is even, then the radicand a must be nonnegative. is not a real number
42 Simplified Form for Radicals of Index n A radical expression of index n is in Simplified Radical Form if it has 1.No perfect nth powers as factors of the radicand, 2.No fractions inside the radical, and 3.No radicals in the denominator.
154 Complex Numbers 1.Definition of i: i =, i 2 = -1. 2.Complex number form: a + bi. 3. a + 0i is the real number a. 4. b is a positive real number 5.The numbers a + bi and a - bi are complex conjugates. Their product is a 2 + b 2. 6.Add, subtract, and multiply complex numbers as if they were algebraic expressions with i being the variable, and replace i 2 by -1. 7.Divide complex numbers by multiplying the numerator and denominator by the conjugate of the denominator. 8.In the complex number system x 2 = k for any real number k is equivalent to