1. Exercise
Evaluate 32. 9

2. Exercise
Evaluate (3 + 4)2. 49

3. Exercise
Evaluate 25

4. Exercise
3 4 2
Evaluate
9 16

5. Exercise
32 42
Evaluate
9 16

6. Square Root
The square root of x is a number whose product when multiplied by itself is x.

7. 92 = 9(9) = 81
(– 9)2 = (– 9)(– 9) = 81
but – 92 = – (9)2 = – 81

8. does not exist in the real number system
√ 81 = 9
– √ 81 = – 9
√ – 81
does not exist in the real number system

9. Example 1
Find √ 25 . 5

10. Example 1
Find – √ 49 .
– 7

11. Example
State whether √ 144 is an integer, real, or not real.
integer

12. Example
State whether √ 132 is an integer, real, or not real.
real

13. Example State whether √ – 36 is an integer, real, or not real.

14. These are called “perfect squares.”
12 = 1
62 = 36
112 = 121
162 = 256
22 = 4
72 = 49
122 = 144
172 = 289
32 = 9
82 = 64
132 = 169
182 = 324
42 = 16
92 = 81
142 = 196
192 = 361
52 = 25
102 = 100
152 = 225
202 = 400
These are called “perfect squares.”

15. 4 1 =
rational number
√ 16
= 4

16. If x is not a perfect square then √ x is an irrational number.

17. √ 5 is …

18. You can estimate √ x by finding the two perfect squares between which x lies.

19. Example 2 Between what two consecutive integers does √ 32 lie?
25 < 32 < 36
√ 25 < √ 32 < √ 36
5 < √ 32 < 6
√ 32 is between 5 and 6.

20. Guess and Check Method:
√ 19
16 < 19 < 25
√ 16 < √ 19 < √ 25
4 < √ 19 < 5
√ 19 ≈ 4.2
Estimate
4.22 = 17.64
√ 19 ≈ 4.3
Estimate
4.32 = 18.49
√ 19 ≈ 4.4
Estimate
4.42 = 19.36
4.3 < √ 19 < 4.4 and √ 19 ≈ 4.4

21. Example 3 Estimate √ 47 to the nearest tenth. 36 < 47 < 49
√ 36 < √ 47 < √ 49
6 < √ 47 < 7

22. Estimate √ 47 to the nearest tenth.
6.92 = 47.61
0.61 greater than 47
6.82 = 46.24
0.76 less than 47
√ 47 ≈ 6.9

23. Example
Order the following numbers from smallest to largest using the < symbol: 11.7, 11.5, √ 135, √ 139.
11.5 < √ 135 < 11.7 < √ 139

24. Example
Determine what integers a2 lies between if a is any single-digit integer.
0 and 100

25. Example
Determine what integers a2 lies between if a is any two-digit integer.
100 and 10,000

26. Example
Estimate √ 53 by finding the two consecutive integers it lies between.
7 and 8

27. Example
Estimate √ 53 by finding its decimal approximation to the nearest integer. 7

28. Example
Estimate √ 53 by finding its decimal approximation to the nearest tenth.
7.3

29. Example
Estimate √ 53 by finding its decimal approximation to the nearest hundredth.
7.28

30. Simplify each square root before adding.
√ x + √ y
Simplify each square root before adding.

31. Simplify by taking the square root after adding.
√ x + y
Simplify by taking the square root after adding.

32. Example 4
Simplify √ 64 + √ 25.
√ 64 + √ 25
= 8 + 5
= 13

33. Example 4
Simplify √ 72 – 23.
√ 72 – 23
= √ 49
= 7

34. Order of Operations
Symbols of Grouping—evaluate quantities within symbols of grouping first.
Exponents and radicals—evaluate a term with an exponent or square root before performing other operations.

35. Order of Operations
3. Multiplication and division—perform these operations in order from left to right.
4. Addition and subtraction—perform these operations last, in order from left to right.

36. Example 5 Simplify – 4 √ 116 – 35. – 4 √ 116 – 35 = – 4 √ 81 = – 4(9)
= – 36

37. Example
Simplify 2 √ 24

38. Example
Simplify 3 √ 25 – 2 √ 64.
– 1

39. Example
3( √ √ 81)
Simplify
4 √ 100 – 64
5 2

40. Exercise
√ 729 x 4
Simplify
√ 81 – √ 4
54 7