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Introduction to Radicals

Introduction to Radicals. Square Root of a Number. If b 2 = a , then b is a square root of a. Radical Expressions. Finding a root of a number is the inverse operation of raising a number to a power. radical sign. index. radicand. This symbol is the radical or the radical sign.

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Introduction to Radicals

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  1. Introduction to Radicals

  2. Square Root of a Number If b 2 = a, then b is a square root of a.

  3. Radical Expressions Finding a root of a number is the inverse operation of raising a number to a power. radical sign index radicand This symbol is the radical or the radical sign • The expression under the radical sign is the radicand. • The index defines the root to be taken.

  4. Terminology: • square root: one of two equal factors of a given number. The radicand is like the “area” of a square and the simplified answer is the length of the side of the squares. • Principal square root: the positive square root of a number; the principal square root of 9 is 3. • negative square root: the negative square root of 9 is –3 and is shown like • radical: the symbol which is read “the square root of a” is called a radical. • radicand: the number or expression inside a radical symbol --- 3 is the radicand. • perfect square: a number that is the square of an integer. 1, 4, 9, 16, 25, 36, etc… are perfect squares.

  5. Square Roots A square root of any positive number has two roots – one is positive and the other is negative. If a is a positive number, then is the positive (principal) square root of aand is the negative square root of a. Examples: non-real #

  6. What does the following symbol represent? The symbol represents the positive or principal root of a number. What is the radicand of the expression ? 5xy

  7. What does the following symbol represent? The symbol represents the negative root of a number. What is the index of the expression ? 3

  8. What numbers are perfect squares? 1 • 1 = 1 2 • 2 = 4 3 • 3 = 9 4 • 4 = 16 5 • 5 = 25 6 • 6 = 36 49, 64, 81, 100, 121, 144, ...

  9. Perfect Squares 64 225 1 81 256 4 100 289 9 121 16 324 144 25 400 169 36 196 49 625

  10. Simplify = 2 = 4 = 5 This is a piece of cake! = 10 = 12

  11. Simplifying Radicals

  12. Simplifying Radical Expressions

  13. Simplifying Radical Expressions • A radical has been simplified when its radicand contains no perfect square factors. • Test to see if it can be divided by 4, then 9, then 25, then 49, etc. • Sometimes factoring the radicand using the “tree” is helpful.

  14. Perfect Square Factor * Other Factor Simplify = = = = LEAVE IN RADICAL FORM = = = = = =

  15. Perfect Square Factor * Other Factor Simplify = = = = LEAVE IN RADICAL FORM = = = = = =

  16. Steps to Simplify Radicals: • Try to divide the radicand into a perfect square for numbers • If there is an exponent make it even by using rules of exponents • Separate the factors to its own square root • Simplify

  17. Simplify: Square root of a variable to an even power = the variable to one-half the power.

  18. Simplify: Square root of a variable to an even power = the variable to one-half the power.

  19. Simplify:

  20. Simplify:

  21. Simplify • . • . • . • .

  22. Simplify • 3x6 • 3x18 • 9x6 • 9x18

  23. Combining Radicals + To combine radicals: combine the coefficients of like radicals

  24. Simplify each expression

  25. Simplify each expression: Simplify each radical first and then combine.

  26. Simplify each expression: Simplify each radical first and then combine.

  27. Perfect Square Factor * Other Factor Simplify = = = = LEAVE IN RADICAL FORM = = = = = =

  28. Simplify each expression

  29. Simplify each expression

  30. Homework radicals 1 • Complete problems 1-24 EVEN from worksheet

  31. Multiplying Radicals * To multiply radicals: multiply the coefficients and then multiply the radicands and then simplify the remaining radicals.

  32. Multiply and then simplify

  33. Dividing Radicals To divide radicals: divide the coefficients, divide the radicands if possible, and rationalize the denominator so that no radical remains in the denominator

  34. That was easy!

  35. This cannot be divided which leaves the radical in the denominator. We do not leave radicals in the denominator. So we need to rationalize by multiplying the fraction by something so we can eliminate the radical in the denominator. 42 cannot be simplified, so we are finished.

  36. This can be divided which leaves the radical in the denominator. We do not leave radicals in the denominator. So we need to rationalize by multiplying the fraction by something so we can eliminate the radical in the denominator.

  37. This cannot be divided which leaves the radical in the denominator. We do not leave radicals in the denominator. So we need to rationalize by multiplying the fraction by something so we can eliminate the radical in the denominator. Reduce the fraction.

  38. Simplify = X = Y3 = P2X3Y = 2X2Y = 5C4D5

  39. Simplify = = = =

  40. Classwork: Packet in Yellow Folder under the desk --- 2nd page Homework: worksheet --- Non-Perfect Squares (#1-12)

  41. Homework radicals 2 • Complete problems 1-15 from worksheet.

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