ARE YOU A RADICAL OR JUST A SQUARE ROOT? EXAMPLES 1. Squaring a number means using that number as a factor two times. 8 8(8) 64 (-8) (-8)(-8) 64 Make sure students realize that x means (x ), not (-x). Thus, parentheses are necessary for indicating the square of a negative number.. Warm-up activity: Have students evaluate each expression: a. 4 b. (-4) c. (0.5) d. (-0.5) ANSWERS a. 16 b. 16 c. 0.5 d. 0.5 3. The inverse of squaring is finding a square root. To find a square root of 64, you must find two equal factors whose product is 64. x x(x) 64 Since 8 times 8 is 64, one square root of 64 is 8. Since (-8) times (-8) is also 64, another square root of 64 is (-8). 4. If x y, the x is a square root of y. Definition of Square Root 0, 1 5. Thought Provoker: What numbers equal their own square roots? 6. An expression like 64 is called a radical expression. The symbol is a radical sign. It indicates the nonnegative or principal square root of the expression under the radical sign called the radicand. Radical sign 64 radicand
7. The square root of a negative number is not defined for this set of numbers (Explain that the square root of a negative number is not defined over the set of real numbers because the square of a real number is never negative. ) Consider the equation 5 n. An equivalent equation is (-5) n. There is no real number value for n that satisfies this equation. 8. Radical sign 64 radicand a) 64 8 64 indicates the principal square root of 64. b) 64-8 64 indicates the negative square root of 64. c) ± 64 ± 8 ± 64 indicates both square roots of 64. ± means positive or negative. 9. Example: Find 49 Since 7 49, we know that 49 7. 10. Example: Find Since 9, we know that 9. 11. Example: Find 11 Since 11 11, we know that 11 11. 1. Example: Find 144 Since 1 144, we know that 144 1. 13. Example: Find - 36 Since (6) 36, we know that - 36-6. 14. Example: Find 5 Since (5) 5, we know that 5-5. 15. Example: Find ± 0. 09 Since (0.3) 0.09, we know that ± 0. 09 ± 0.3. 16. Example: Find ± 1. 69 Since (1.3) 1.69, we know that ± 1. 69 ± 1. 3
17. To simplify a radical expression, find the square root of any factors of the radicand that are perfect squares. For example, to simplify 196, find any factors of 196 that are perfect squares. You can use the prime factorization of 196 shown: Check: 14 196 Encourage students to check square roots by squaring! 196 7 7 7 7 7 14 18. For any numbers a and b, where a 0 and b 0, ab a b. Product Property of Square Roots 19. Before introducing the next example, have students find square roots of numbers when given the prime factorizations. Point out that to find the square root of a power, divide the exponent by. For example: 4 4 3 5 (3 )(5) 180 0. Example: Simplify 576. Find the prime factorization of 576. Express the radicand using exponents. Use the Product Property of Square Roots. Simplify each radical. 576 3 3 6 3 6 3 3 4 3 Check: 4 576
1. A similar property for quotients can also be used to simply square roots. For any numbers a and b, where a 0 and b > 0, a a b b Quotient Property of Square Roots. Example: Simplify 9 16 9 9 3 Check: 16 16 4 3 9 ( ) 16 4 3. Example: Simplify 11 11 11 9 Check: 11 9 11 ( ) 11
Name: Date: Class: ARE YOU A RADICAL OR JUST A SQUARE ROOT? WORKSHEET State the square of each number: 1. 10. 1 3. 7 4. 0 5. 0.3 7. 8. 9. 1 4 7 7 8 6. 0.04 10. 11 4 Find the principal square root of each number: 11. 49 1. 64 13. 16 14. 169 5 15. 36 16. 11 17. 0.0009 18. 0.005 19. 0.0036 0. 0.16 Simplify: 1. 36. 100 5. 89 100 3. ± 144 4. 0. 36 6. ± 961 79
ARE YOU A RADICAL OR JUST A SQUARE ROOT? WORKSHEET KEY State the square of each number: 1. 10 100. 1 144 3. 7 49 4. 0 400 5. 0.3 0.09 7. 8. 9. 1 1 4 4 16 7 49 7 49 8 64 6. 0.04.0016 10. 11 11 4 16 Find the principal square root of each number: 11. 49 7 1. 64 8 13. 16 4 14. 169 13 5 5 15. 36 6 9 16. 11 11 17. 0.0009 0.03 18. 0.005 0.05 19. 0.0036 0.06 0. 0.16 0.4 Simplify: 1. 36 6. 100-10 3. ± 144 ± 1 4. 0. 36 0.6 5. 6. 89 100 961 ± 79 17 10 31 ± 7
Student Name: Date: ARE YOU A RADICAL OR JUST A SQUARE ROOT? CHECKLIST 1. On questions 1 thru 10, did the student state the square of each number correctly? a. All ten (50 b. Nine of the ten (45 c. Eight of the ten (40 d. Seven of the ten (35 e. Six of the ten (30 f. Five of the ten (5 g. Four of the ten (0 h. Three of the ten (15 i. Two of the ten (10 j. One of the ten (5. On questions 11 thru 0, did the student find the principal square root of each number correctly? a. All ten (50 b. Nine of the ten (45 c. Eight of the ten (40 d. Seven of the ten (35 e. Six of the ten (30 f. Five of the ten (5 g. Four of the ten (0 h. Three of the ten (15 i. Two of the ten (10 j. One of the ten (5 3. On questions 1 thru 6, did the student simplify the radical correctly? a. All six (30 b. Five of the six (5 c. Four of the six (0 d. Three of the six (15 e. Two of the six (10 f. One of the six (5 Total Number of Points A B C D F 117 points and above 104 points and above 91 points and above 78 points and above 77 points and below Any score below C needs remediation!