1. 1 Topic 1.3.2 Square Roots

2. 2 Topic 1.3.2 Square Roots California Standard: 2.0: Students understand and use such operations as taking the opposite, finding the reciprocal, taking a root, and raising to a fractional power. They understand and use the rules of exponents. What it means for you: You’ll look more closely at the rules of square roots. Key words: square root radical radicand principal square root minor square root

3. 3 Topic 1.3.2 Square Roots In the last Topic you learned about all the exponent rules — this Topic will look more closely at one rule in particular. Square roots are the type of root that you’ll come across most often in math problems — so it’s really important that you know how to deal with them.

4. 4 Topic 1.3.2 Another Name for the Root Sign is the Radical Sign Square Roots The square root of p is written. If you multiply by itself, you get p — so × = p. Multiplying by itself means you square it. If you raise to the power n, you get p — so = p. n The symbol is called the radical sign and shows the nonnegative root if more than one root exists. In the expression (the n th root of p ), p is the radicand. The n th root of p is written.

5. 5 Topic 1.3.2 Square Roots The square root of a number p is also written p. 1 2 For any real number p > 0, the square root is written as (or p ). 1 2 If r =, then r 2 = p and (– r ) 2 = p. r is called the principal square root of p and – r is called the minor square root of p. You can show this using the rules of exponents: 2 1 2 2 2 1 p = p = p = p

6. 6 2. The 6 th root of t is written in radical notation. …….. Topic 1.3.2 Guided Practice Solution follows… Square Roots Complete the following. 1. The radicand of 8 is. …….. 3 3. 9 × = 9 …….. 4. b = in radical notation. 1 2 …….. 8 t 6 9 b

7. 7 Topic 1.3.2 Positive Numbers Have Two Square Roots Square Roots Every positive number has two square roots — a positive one (the principal square root) and a negative one (the minor square root). The principal square root of n is written as. The minor square root of n is written as –. To indicate both square roots you can write ±.

8. 8 Topic 1.3.2 Example 1 Solution follows… Square Roots Find the square roots of the following numbers: a) 100 b) n 2 Solution a) 100 = 10 So the principal square root is 10, and the minor square root is –10. b) n 2 = | n | So the principal square root is | n |, and the minor square root is –| n |.

9. 9 Topic 1.3.2 Guided Practice Solution follows… Square Roots Find the principal square root and minor square root of these numbers: 5. 46. 1007. 81 Use the “±” symbol to give the principal and minor square root of the following numbers: 8. 99. 1610. 144 11. 35 2 12. x 2 13. 81 14. t 2 15. 9 × 916. ( st ) 2 Evaluate the following, giving the principal and minor roots: 17. 4 1 2 18. 121 1 2 2 and –210 and –109 and –9 ±3±4±12 ±35±x±x ±9 ±t±t ± st ±2±11

10. 10 Topic 1.3.2 Algebraic Expressions Also Have Square Roots Square Roots You can also take the square root of an algebraic expression. Remember — algebraic expressions contain variables, which represent unknown values. For example, a + b or 2 t 4.

11. 11 Topic 1.3.2 Example 2 Solution follows… Square Roots Find the square root of ( x + 1) 2. Solution ( x + 1) 2 = | x + 1| So the principal square root is | x + 1| and the minor square root is –| x + 1|.

12. 12 Topic 1.3.2 Guided Practice Solution follows… Square Roots Give the principal and minor square root of each of the following expressions. 19. t × t ±t±t 20. t 2 × t 2 ±t2±t2 21. a 2 × a 2 ±a2±a2 22. ( a + b ) × ( a + b ) ±( a + b ) 23. t ( a + b ) × t ( a + b ) ± t ( a + b ) 24. ( a + b ) 2 ±( a + b ) 25. ( t + 1) 2 ±( t + 1) 26. [ t ( a + b )] 2 ± t ( a + b ) 27. [2( a + b )] 2 ±2( a + b )

13. 13 Topic 1.3.2 Independent Practice Solution follows… Square Roots 1. Is this statement true or false? “The radicand of 32 is 5.” 5 Evaluate the following. 2. 64 1 2 3. (49)4. a 2 5. 256. 12 2 7. j × j Find the square roots of the following. 8. ( a 2 ) 2 9. ( k – 1) 2 10. ( m + n ) 2 11. ( m 2 + n 2 ) 2 12. (2 pq ) 2 13. [( a + b ) × ( c + d )] 2 False. The radicand is 32. 87|a||a| 512|j||j| ±a2±a2 ±( k – 1)±( m + n ) ±( m 2 + n 2 )±2 pq ±( a + b )( c + d )

14. 14 Topic 1.3.2 Round Up Square Roots Remember that when you take the square root of a positive number, you always have two possible answers — a positive one and a negative one. You can give both answers neatly using the ± sign.