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Understanding Square Roots The square root of a given number is a value that, when multiplied by itself, produces that given number. Square roots come in pairs, a positive root and a negative root. For example, a square root of 4 is 2, because 2 × 2 = 4. Another square root of 4 is −2, because −2 × −2 = 4. This can be written as 4 = ±2, which means that 4 = 2 and 4 = −2. Note: You can not take the square root of a negative number. To better understand square roots, it is very helpful to be familiar with perfect squares. A perfect square is the square of an integer: example 1 Which of the following statements about 121 is not true? A. 121 is an irrational number. B. 121 is an integer. C. 121 is a real number. D. 121 is a rational number. To answer this question, knowledge of perfect squares is extremely helpful. A square root is an irrational number, unless it is the square root of a perfect square: 11 is an integer and a real number and a rational number, but 11 is not an irrational number, so 121 is not an irrational number. The answer is A. Simplifying Square Roots To simplify the square root of a number, find two factors of that number, one of which is a perfect square. (A factor is an integer that divides evenly into another number.) example 2 Simplify 18 . This example demonstrates the following property of square roots: This applies to fractions as well: example 3 Simplify 4 . 9 example 4 Simplify 32 . 20 It’s not acceptable to leave a square root (also called a radical) in the denominator of a fraction. The way to leave an answer without a radical in the denominator is to multiply the numerator and denominator by the same radical that is in the denominator. This is called rationalizing the denominator. Estimating the Value of a Square Root Perfect squares can also be used to estimate the approximate value of a square root. example 5 Which is the best approximation of A. B. C. D. 72 ? 7.2 9.1 8.9 8.5 example 6 The square root of 31 is between which two whole numbers? A. B. C. D. 4 and 5 5 and 6 6 and 7 7 and 8 This problem can also be solved by first making a list of perfect squares. 31 is between 25 and 36, so the square root of 31 is between 25 and 36 , or between 5 and 6. The answer is B. example 7 A part of the real number line is shown below. Q 0 5 R S 10 15 T 20 25 Which letter best represents the location of 50 ? A. B. C. D. Q R S T Once again, a list of perfect squares will help, this time near 50. 50 is really close to 49, so 50 will be close to 49 , or 7. On the number line above, only one point looks like it could represent a value near 7. The answer is B. Name ______________________________ Simplify. Leave in radical form (in other words, don’t use a calculator). 1) 20 2) − 64 3) 900 4) 0.16 5) − 0.0025 6) 16 25 7) 3 27 8) 10) 8 · 32 4 3 11) 5 14 · 2 7 9) 3 7 12) − 3 5 · 4 5 Name the integers between which each value lies (without using a calculator). 13) 11 15) 2 3 17) 25 2 14 ) 34 16 ) 3 6 18) 14 2 Label each statement as true or false (without using a calculator). 19) 2 < 7 < 3 20) 62 < 65 < 72 21) 22 < 20 < 32 22) 45 < 10 < 90