Radical Expressions and Functions What is a square root of 25? How many square roots does 25 have? Do the following square roots exist?

Transcription

1 Hartfield Intermediate Algebra (Version D) Unit 4 Page 1 Topic 4 1 Radical Epressions and Functions What is a square root of 25? How many square roots does 25 have? Do the following square roots eist? 4 4 Definition: X is a square root of a if X² = a. 0 Symbolically, a is the principle square root of a. To symbolically represent each square root of a, one must write a and a. This leads to the short hand way of writing both square roots as a. 4

2 Hartfield Intermediate Algebra (Version D) Unit 4 Page 2 What are the following square roots? z 4 In general. is called a radical sign or a root sign. A square root is a particular type of root that uses the root sign for itself. 4 64z is an eample of a radical epression since it an epression with a root sign. In the above epression, the 64z is the radicand. The radicand is the epression under (or better said, inside) a radical epression. f() is an eample of a radical function. 4

3 Hartfield Intermediate Algebra (Version D) Unit 4 Page Definition: A number S is called a perfect square if it s the result of squaring an integer. You need to memorize the first 21 numeric perfect squares Variable epressions can be perfect squares also if we amend the definition as follows: An epression is a perfect square if its coefficient satisfies the definition of a numeric perfect square & each variable has an integer eponent that is a multiple of 2. The square root of a numeric value that isn t a perfect square usually results in an irrational number. Recall that irrational numbers cannot be epressed as fractions of integers and their decimal form neither repeats nor terminates.

4 Hartfield Intermediate Algebra (Version D) Unit 4 Page 4 Definition: X is a cube root of a if X³ = a. a X X a All numbers have one cube root thus every cube root is a principle cube root Definition: A number C is called a perfect cube if it s the result of cubing an integer. You need to memorize the first 11 numeric perfect cubes Variable epressions can be perfect cubes also if we amend the definition as follows: An epression is a perfect cube if its coefficient satisfies the definition of a numeric perfect cube & each variable has an integer eponent that is a multiple of.

5 Hartfield Intermediate Algebra (Version D) Unit 4 Page 5 Definitions: X is a fourth root of a if X 4 = a. X is a fifth root of a if X 5 = a. X is an nth root of a if X n = a. All roots have an inde. The inde of a root is equal to the power needed to return X to a by the previously state definitions. Roots with an even inde (such as square roots and fourth roots) Positive number have 2 real roots. Zero is its own root. Negative numbers have 0 real roots. Roots with an odd inde (such as cube roots and fifth roots) All numbers have eactly one real root. Notationally write the fourth roots of 81 and evaluate. Notationally write the fifth root of 24 and evaluate. Definitions: A number R is called a perfect fourth if it s the result of raising an integer to a fourth power. A number R is called a perfect fifth if it s the result of raising an integer to a fifth power.

6 Hartfield Intermediate Algebra (Version D) Unit 4 Page 6 You need to memorize the first 6 numeric perfect fourths and first 5 numeric perfect fifths. Perfect fourths: Perfect fifths: Find each root. Assume that all variables represent non negative real numbers For roots with even indices, keep in mind the following rule: If variables can represent any real number, you may need to use absolute value symbols when simplifying. If the variables can only represent non negative numbers, you won t need absolute value symbols when simplifying. Absolute value symbols are never needed if a root has an odd inde y yz

7 Hartfield Intermediate Algebra (Version D) Unit 4 Page 7 Find each root. Assume that all variables can represent any real number. Find each root. Assume that all variables can represent any real number y

8 Hartfield Intermediate Algebra (Version D) Unit 4 Page 8 Topic 4 2 Radicals and Rational Eponents Recall the Laws of Eponents ( > 0) a b ab hw 1 Think about how the Laws of Eponents are related here: a b ab a n an 1 Also: m m m y y and y m y m m n 1 n

9 Hartfield Intermediate Algebra (Version D) Unit 4 Page 9 Eplore the possibilities associated with the following rational eponent: Evaluate and/or simplify. Assume that all variables represent non negative real numbers In general we can conclude that m n n n m m

10 Hartfield Intermediate Algebra (Version D) Unit 4 Page 10 Rewrite each epression in radical notation and simplify as possible. Assume that all variables represent nonnegative real numbers m 1 Recall that define m 1 n. m n Evaluate. m which we can etend to

11 Hartfield Intermediate Algebra (Version D) Unit 4 Page 11 Use the properties of eponents to simplify each epression. Write your final answers with positive eponents. Use the properties of eponents to simplify each epression. Write your final answers with positive eponents

12 Hartfield Intermediate Algebra (Version D) Unit 4 Page 12 Multiply. hw 2 Factor

13 Hartfield Intermediate Algebra (Version D) Unit 4 Page 1 Use rational eponents with each to find a single simplified radical. Assume that all variables represent non negative real numbers. Use rational eponents with each to find a single simplified radical. Assume that all variables represent non negative real numbers y

14 Hartfield Intermediate Algebra (Version D) Unit 4 Page 14 Topic 4 Product/Quotient Rules and Simplifying n n n Product Rule for Radicals: a b a b Multiply. Assume that all variables represent nonnegative real numbers. Quotient Rule for Radicals: n n a b n a b Divide

15 Hartfield Intermediate Algebra (Version D) Unit 4 Page 15 To simplify radicals, apply the product and quotient rules in reverse. Simplify Simplify

16 Hartfield Intermediate Algebra (Version D) Unit 4 Page 16 Simplify Emphasis: it s all about perfect squares, cubes, etc. Simplify. Assume that all variables represent nonnegative real numbers. 5 10

17 Hartfield Intermediate Algebra (Version D) Unit 4 Page 17 Simplify. Assume that all variables represent nonnegative real numbers. Simplify. Assume that all variables represent nonnegative real numbers y yz yz

18 Hartfield Intermediate Algebra (Version D) Unit 4 Page 18 Simplify. Assume that all variables represent nonnegative real numbers. Simplify. Assume that all variables represent nonnegative real numbers y

19 Hartfield Intermediate Algebra (Version D) Unit 4 Page 19 Topic 4 4 Adding and Subtracting Radicals Compare the following pairs of sums 4y 24 Add To add or subtract radicals, you must have like radicals. Like radicals have the same radicand and the same root inde.

20 Hartfield Intermediate Algebra (Version D) Unit 4 Page 20 Subtract Subtract

21 Hartfield Intermediate Algebra (Version D) Unit 4 Page 21 Add and/or subtract Add. Assume that all variables represent non negative real numbers

22 Hartfield Intermediate Algebra (Version D) Unit 4 Page 22 Add

23 Hartfield Intermediate Algebra (Version D) Unit 4 Page 2 Topic 4 5 More Multiplying Radicals To multiply radicals with coefficients, keep the following rule in mind: Multiply Multiply. n n n ay b y a b

24 Hartfield Intermediate Algebra (Version D) Unit 4 Page 24 Multiply Multiply

25 Hartfield Intermediate Algebra (Version D) Unit 4 Page 25 Multiply

26 Hartfield Intermediate Algebra (Version D) Unit 4 Page 26 Topic 4 6 Rationalizing Radical Epressions Imagine if you had to divide the following epressions, which would be easier? (Note, ) 2 2 Rationalizing the denominator of a fraction: to rewrite a fraction in an equivalent form where no radical is present in the denominator. There are three cases that vary the technique of rationalizing the denominator, based on what is in the denominator. Traditionally, rationalizing the denominator of a radical epression was done for computational purposes. Today, it is used less frequently but is still a useful skill.

27 Hartfield Intermediate Algebra (Version D) Unit 4 Page 27 Case 1: The denominator is a square root. Rationalize the denominator of each epression. Rationalize the denominator of the epression To rationalize the denominator when it is a square root, simplify the denominator and multiply by an identity fraction involving only the radical part of the simplified radical.

28 Hartfield Intermediate Algebra (Version D) Unit 4 Page 28 Rationalize the denominator of each epression Case 2: The denominator is a cube root, fourth root, or any other root. Rationalize the denominator of the epression. 5 4 To rationalize the denominator when it is any root other than a square root, simplify the denominator, then determine the smaller perfect cube (fourth, etc) that the radicand of the denominator will divide. Create an identity fraction using an appropriate radical to create the perfect number under the root.

29 Hartfield Intermediate Algebra (Version D) Unit 4 Page 29 Rationalize the denominator of each epression. Rationalize the denominator of each epression

30 Hartfield Intermediate Algebra (Version D) Unit 4 Page 0 Case : The denominator is a square root ± a number or another root. Rationalize the denominator of the epression. Rationalize the denominator of each epression To rationalize the denominator when it consists of a square root plus or minus a number or another root, create an identity fraction using the conjugate pair of the denominator.

31 Hartfield Intermediate Algebra (Version D) Unit 4 Page 1 Rationalize the denominator of each epression. 4 7 Rationalize the denominator of each epression