Topic 1.3.2

1. Square Roots Topic 1.3.2

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

5. 1 1 2 1 The square root of a number p is also written p . 2 2 2 2 You can show this using the rules of exponents: 2 1 p = p = p = p For any real number p > 0, the square root is written as (or p ). If r = , then r2 = p and (–r)2 = p. Topic 1.3.2 Square Roots r is called the principal square root of p and –r is called the minor square root of p.

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

7. 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 ± . Topic 1.3.2 Square Roots Positive Numbers Have Two Square Roots Every positive number has two square roots — a positive one (the principal square root) and a negative one (the minor square root).

8. a) 100 = 10 b) n2 = |n| Topic 1.3.2 Square Roots Example 1 Find the square roots of the following numbers: a) 100 b) n2 Solution So the principal square root is 10, and the minor square root is –10. So the principal square root is |n|, and the minor square root is –|n|. Solution follows…

9. 1 1 2 2 17. 4 18. 121 Topic 1.3.2 Square Roots Guided Practice Find the principal square root and minor square root of these numbers: 5. 4 6. 100 7. 81 2 and –2 10 and –10 9 and –9 Use the “±” symbol to give the principal and minor square root of the following numbers: 8. 9 9. 16 10. 144 ±3 ±4 ±12 11. 352 12.x2 13. 81 ±35 ±x ±9 14.t2 15. 9 × 9 16. (st)2 ±t ±9 ±st Evaluate the following, giving the principal and minor roots: ±2 ±11 Solution follows…

10. Remember — algebraic expressions contain variables, which represent unknown values. For example, a + b or 2t4. Topic 1.3.2 Square Roots Algebraic Expressions Also Have Square Roots You can also take the square root of an algebraic expression.

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

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

13. 1 2 Topic 1.3.2 Square Roots Independent Practice 5 1. Is this statement true or false? “The radicand of 32 is 5.” False. The radicand is 32. Evaluate the following. 2. 64 3. (49) 4. a2 8 7 |a| 5. 25 6. 122 7. j × j 5 12 |j| Find the square roots of the following. 8. (a2)2 9. (k – 1)2 10. (m + n)2 ±a2 ±(k – 1) ±(m + n) 11. (m2 + n2)2 12. (2pq)2 13. [(a + b) × (c + d)]2 ±(m2+ n2) ±2pq ±(a + b)(c + d) Solution follows…

14. Topic 1.3.2 Square Roots Round Up 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.