INTEGERS, POWERS AND ROOTS 1.4 Powers and roots Powers are used when we want to multiply a number by itself repeatedly. A number multiplied by itself several times is called a power of that number. Use indices to show powers. For the example 52, we say that: 5 is the base and 2 is the index (or power, or exponent). 52 means "multiply 5 by itself 2 times". ■ 52 = 5 × 5 = 25 This is five squared or the square of five. ■ 53 = 5 × 5 × 5 = 125 This is five cubed or the cube of five. ■ 54 = 5 × 5 × 5 × 5 = 625 This is five to the power four. ■ 55 = 5 × 5 × 5 × 5 × 5 = 3125 This is five to the power five. Roots are the opposite of powers. The square of 5 is 52 = 25. Therefore, the square root of 25 is 5 and you write this as 25 The cube of 5 is 53 = 125. Therefore, the cube root of 125 is 5 and you write this as 3 125. Square numbers have square roots that are integers. Examples: 132 = 169, so 169= 13 192 = 361, so 361= 19 ■ 9 (square root of 9) has 2 answers 9 = ퟑ 푎푛푑 − 9 = −ퟑ We know that 3×3 = 32 = 9, So reversing it gives : 9=3 . Also notice: −3×−3 = (−3)2 = 9 so also 9 = −3 This is often written as 9 = ±ퟑ (meaning +3 or -3) ■ 3 27 (cubed root of 27) = 3 We know: 3 x 3 x 3 = 33 = 27, therefore 3 27= 3 Also notice: −3 × −3 × −3 = (−3)3 = −27 so there is only 1 cubed root. How to Calculate a Square Root Using Prime Factorization 1. Divide your number into perfect square factors. This method uses a number's factors to find a number's square root (depending on the number, this can be an exact numerical answer or a close estimate). To start finding a square root via prime factorization, first, try to reduce your number into its perfect square factors 2. Take the square roots of your perfect square factors. The product property of square roots states that for any given numbers a and b, 푎 푥 푏 = 푎 푥 푏. Because of this property, we can now take the square roots of our perfect square factors and multiply them together to get our answer. Example : Find the square root of 400 by hand.