Hartfield Intermediate Algebra (Version 2014-2D) 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
Hartfield Intermediate Algebra (Version 2014-2D) Unit 4 Page 2 What are the following square roots? 9 49 0.0016 8 64z 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
Hartfield Intermediate Algebra (Version 2014-2D) 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. 0 1 121 4 144 9 169 16 196 25 225 6 256 49 289 64 24 81 61 100 400 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.
Hartfield Intermediate Algebra (Version 2014-2D) 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. 64 64 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. 0 1 8 27 64 125 216 4 512 729 1000 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.
Hartfield Intermediate Algebra (Version 2014-2D) 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.
Hartfield Intermediate Algebra (Version 2014-2D) Unit 4 Page 6 You need to memorize the first 6 numeric perfect fourths and first 5 numeric perfect fifths. Perfect fourths: 0 1 16 81 256 625 Perfect fifths: 0 1 2 24 1024 Find each root. Assume that all variables represent non negative real numbers. 4 4 625 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. 9 12 125 y 64 12 2 6 yz
Hartfield Intermediate Algebra (Version 2014-2D) 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. 4 4 625 27 4 4 12 y 4 4 625 4 8 27 5 20 2 2 6 9
Hartfield Intermediate Algebra (Version 2014-2D) 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: 2 1 2 a b ab a n an 1 Also: m m m y y and y m y m m n 1 n
Hartfield Intermediate Algebra (Version 2014-2D) 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. 4 49 1 2 8 2 In general we can conclude that m n n n m m. 9 64 1 8 2 1 12
Hartfield Intermediate Algebra (Version 2014-2D) Unit 4 Page 10 Rewrite each epression in radical notation and simplify as possible. Assume that all variables represent nonnegative real numbers. 27 2 9 m 1 Recall that define m 1 n. m n Evaluate. m which we can etend to 5 1 4 2 27 1 7 5 16 4 81 2 4 6 2
Hartfield Intermediate Algebra (Version 2014-2D) 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. 4 1 2 5 6 4 2 4 2 1 2 1 5 4 1 10
Hartfield Intermediate Algebra (Version 2014-2D) Unit 4 Page 12 Multiply. hw 2 Factor. 2 1 1 2 2 4 5 5 1 1 1 2 2 6 1 5 7 7 2
Hartfield Intermediate Algebra (Version 2014-2D) 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. 4 2 4 6 6 9 y 5 2 8 6 4
Hartfield Intermediate Algebra (Version 2014-2D) 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 7 5 9 Divide. 15 54 2 2 8 4 2
Hartfield Intermediate Algebra (Version 2014-2D) Unit 4 Page 15 To simplify radicals, apply the product and quotient rules in reverse. Simplify. 75 162 Simplify. 48 128
Hartfield Intermediate Algebra (Version 2014-2D) Unit 4 Page 16 Simplify. 4 48 56 Emphasis: it s all about perfect squares, cubes, etc. Simplify. Assume that all variables represent nonnegative real numbers. 5 10
Hartfield Intermediate Algebra (Version 2014-2D) Unit 4 Page 17 Simplify. Assume that all variables represent nonnegative real numbers. Simplify. Assume that all variables represent nonnegative real numbers. 18 8 24y 5 8 11 50 yz 7 2 6 40 yz
Hartfield Intermediate Algebra (Version 2014-2D) Unit 4 Page 18 Simplify. Assume that all variables represent nonnegative real numbers. Simplify. Assume that all variables represent nonnegative real numbers. 6 25 2 y 49 4 12
Hartfield Intermediate Algebra (Version 2014-2D) Unit 4 Page 19 Topic 4 4 Adding and Subtracting Radicals Compare the following pairs of sums 4y 24 Add. 8 27 5 12 2 4 24 2 4 24 2 To add or subtract radicals, you must have like radicals. Like radicals have the same radicand and the same root inde.
Hartfield Intermediate Algebra (Version 2014-2D) Unit 4 Page 20 Subtract. 80 2 45 Subtract. 5 2 108
Hartfield Intermediate Algebra (Version 2014-2D) Unit 4 Page 21 Add and/or subtract. 4 50 245 2 54 Add. Assume that all variables represent non negative real numbers. 2 28 7
Hartfield Intermediate Algebra (Version 2014-2D) Unit 4 Page 22 Add. 5 112 28 9 9
Hartfield Intermediate Algebra (Version 2014-2D) Unit 4 Page 2 Topic 4 5 More Multiplying Radicals To multiply radicals with coefficients, keep the following rule in mind: Multiply. 2 6 5 7 4 Multiply. n n n ay b y a b 2 4 5 2
Hartfield Intermediate Algebra (Version 2014-2D) Unit 4 Page 24 Multiply. 7 2 4 6 2 Multiply. 2 6 22 6 2
Hartfield Intermediate Algebra (Version 2014-2D) Unit 4 Page 25 Multiply. 4 2 2
Hartfield Intermediate Algebra (Version 2014-2D) Unit 4 Page 26 Topic 4 6 Rationalizing Radical Epressions Imagine if you had to divide the following epressions, which would be easier? (Note, 1.720508) 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.
Hartfield Intermediate Algebra (Version 2014-2D) Unit 4 Page 27 Case 1: The denominator is a square root. Rationalize the denominator of each epression. Rationalize the denominator of the epression. 6 2 5 12 27 50 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.
Hartfield Intermediate Algebra (Version 2014-2D) Unit 4 Page 28 Rationalize the denominator of each epression. 15 6 8 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.
Hartfield Intermediate Algebra (Version 2014-2D) Unit 4 Page 29 Rationalize the denominator of each epression. Rationalize the denominator of each epression. 7 2 2 4 5 2 6 4 9
Hartfield Intermediate Algebra (Version 2014-2D) 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. 4 7 5 1 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.
Hartfield Intermediate Algebra (Version 2014-2D) Unit 4 Page 1 Rationalize the denominator of each epression. 4 7 Rationalize the denominator of each epression. 5 2 5