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radicals

1. Introduction to Radicals

2. Square Root of a Number If b 2 = a , then b is a square root of a . Meaning Positive Square Root Negative Square Root The positive and negative square roots Symbol Example

3. Radical Expressions Finding a root of a number is the inverse operation of raising a number to a power. This symbol is the radical or the radical sign index radical sign radicand 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 If a is a positive number, then is the positive (principal) square root of a and is the negative square root of a . A square root of any positive number has two roots – one is positive and the other is negative. Examples: non-real #

6. In the radicand expression   If a 0, then is zero or a positive real number. If 0 and n is odd, then is a negative real number. If 0 and n is even, then is not a real number.  

7. 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

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

9. Simplifying Radicals

10. Simplifying Radical Expressions

11. 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.

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

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

14. 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

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

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

17. Simplify:

18. Simplify:

19. Simplify . . . .

20. Simplify 3x 6 3x 18 9x 6 9x 18

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

22. Simplify each expression

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

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

25. = = = = = Simplify = = = = = Perfect Square Factor * Other Factor

26. Simplify each expression

27. Simplify each expression

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

29. Multiply and then simplify

30. 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

31. 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.

32. 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.

33. 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.

34. = 15 = 5x =2 1.   2.   3.   Perform the given operation

35. To rationalize the denominator with a binomial radical expression is to multiply the numerator and the denominator of the fraction by the conjugate of the denominator. a.   b.