Algebraic Roots and Radicals

1. Algebraic Roots and Radicals Simplifying Perfect Square Radical Expressions Approximating Square Roots Click on topic to go to that section. Rational & Irrational Numbers Radical Expressions Containing Variables Simplifying Non-Perfect Square Radicands Simplifying Roots of Variables Operations with Radicals Pythagorean Theorem Distance Formula Intro to Trig Solving Right Triangles

2. Simplifying Perfect Square 
Radical Expressions Return to 
Table of 
Contents

3. Can you recall the perfect squares from 1 to 169?  12 =  82 =  22 = 92 =  32 = 102 =  42 =  112 =  52 =  122 =  62 = 132 = 202 =  72 =

4. Square Root Of A Number Recall: If b2 = a, then b is a square root of a. Example: If 42 = 16, then 4 is a square root of 16 What is a square root of 25? 64? 100?

5. Square Root Of A Number Square roots are written with a radical symbol Positive square root: = 4 Negative square root: - = - 4 Positive & negative square roots: = 4 Negative numbers have no real square roots no real roots because there is no real number that, when squared, would equal -16.

6. Is there a difference between & ? Which expression has no real roots? Evaluate the expression

7. Evaluate the expression is not real

8. 1

9. 2 ?

10. 3 = ?

11. 4

12. 5

13. 6 = ? A 3 B -3 C No real roots

14. 7 The expression equal to is equivalent to a positive integer when b is A -10 B 64 C 16 D 4

15. Square Roots of Fractions a b = b0 4 16 49 = = 7

16. Try These

17. 8 C A B D no real solution

18. 9 C A B D no real solution

19. 10 C A B D no real solution

20. 11 C A B D no real solution

21. 12 C A B D no real solution

22. Square Roots of Decimals Recall:

23. To find the square root of a decimal, convert the decimal 
to a fraction first. Follow your steps for square roots of 
fractions. = .2 = .05 = .3

24. 13 Evaluate B A C D No Real Solution

25. 14 Evaluate B .6 A .06 C 6 D No Real Solution

26. 15 Evaluate B 11 A .11 C 1.1 D No Real Solution

27. 16 Evaluate B .08 A .8 C D No Real Solution

28. 17 Evaluate B A C D No Real Solution

29. Approximating Square Roots Return to 
Table of 
Contents

30. Approximating a Square Root Approximate to the nearest integer < Identify perfect squares closest to 38 Take square root < < 6 7 < Answer: Because 38 is closer to 36 than to 49, is closer to 6 
than to 7. So, to the nearest integer, = 6

31. Approximate to the nearest integer < < Identify perfect squares closest to 70 Take square root Identify nearest integer < <

32. 18 Approximate to the nearest integer

33. 19 Approximate to the nearest integer

34. 20 Approximate to the nearest integer

35. 21 Approximate to the nearest integer

36. 22 Approximate to the nearest integer

37. 23 The expression is a number between A 3 and 9 B 8 and 9 C 9 and 10 D 46 and 47

38. Rational & Irrational Numbers Return to 
Table of 
Contents

39. Rational & Irrational Numbers is rational because the radicand (number under the radical) is a perfect square If a radicand is not a perfect square, the root is said to be 
irrational. Ex:

40. Sort by the square root being rational or irrational.

41. 24 Rational or Irrational? A Rational B Irrational

42. 25 Rational or Irrational? A Rational B Irrational

43. 26 Rational or Irrational? A Rational B Irrational

44. 27 Rational or Irrational? A Rational B Irrational

45. 28 Rational or Irrational? A Rational B Irrational

46. 29 Which is a rational number? A B p C D

47. 30 Given the statement: “If x is a rational number, 
then is irrational.”Which value of x makes 
the statement false? A B 2 C 3 D 4

48. Radical Expressions Containing Variables Return to 
Table of 
Contents

49. Square Roots of Variables To take the square root of a variable rewrite its exponent as 
the square of a power. = =

50. Square Roots of Variables If the square root of a variable raised to an even power has a variable raised to an odd power for ananswer, the answer must have absolutevalue signs. This ensures that the answer will be positive. By Definition...