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P.4 Rational Exponents and Radicals Copyright © Cengage Learning. All rights reserved. Objectives ■ Radicals ■ Rational Exponents ■ Rationalizing the Denominator; Standard Form 2 Radicals 3 Radicals We know what 2n means whenever n is an integer. To give meaning to a power, such as 24/5, whose exponent is a rational number, we need to discuss radicals. The symbol means “the positive square root of.” Thus Since a = b2 0, the symbol a 0. For instance, =3 because makes sense only when 32 = 9 and 30 4 Radicals Square roots are special cases of nth roots. The nth root of x is the number that, when raised to the nth power, gives x. 5 Radicals 6 Example 1 – Simplifying Expressions Involving nth Roots (a) Factor out the largest cube Property 1: Property 4: (b) Property 1: Property 5, Property 5: 7 Example 2 – Combining Radicals (a) Factor out the largest squares Property 1 Distributive property (b) If b > 0, then Property 1: = Property 5, b > 0 Distributive Property 8 Example 2 – Combining Radicals (c) cont’d Factor out 49 Property 1: 9 Simplify Radicals This needs to be done on every single problem before you write an answer. The general accepted practice is to never write 2/4 as an answer we simplify to ½, the same applies here. If a radical can be written in a simpler form then we want to do so. When changing to a simpler form you should not impact the actual values represented. When I change 2/4 to ½ my calculator for both gives 0.5 this tells me that the simplification to ½ was right. We can use the same argument for checking our simplified radicals. When you simplify things you change the way it looks not the amount that it represents. If checking yourself on questions with variables you can still use your calculator just make sure that the values you choose for your initial problem are the same values that you use for your check. If you let x=2 in the initial problem you better let x=2 in your simplified version too. 10 11 12 12.2 Simplifying Radicals When simplifying radicals it is best to deal with the numerical parts separate. We will conduct what is called the PRIME FACTORIZATION of the radicand. -That means we will break the radicand down to the factors that multiply together to give the radicand. We will then group them based on the INDEX. 12 72 13 3 48 3 120 14 Now we can do the same thing with variables under the radical. You will want to re-write the variables that have higher exponents in multiples that relate directly to the INDEX 5 For example… 2 x 3 y8 15 Now put it all together inside one problem 16 Simplifying Fractions under a radical Break it up into two separate radicals. Numerator and denominator Then simplify each radical as far as possible. 5 9 25 81 17 When you do this and a radical is left in the denominator you need to do what is called “Rationalizing the denominator”. To do this we multiply by a convenient value of 1. (that value will always be whatever the denominator is) After you multiply you may have to reduce the radicals again. 25 72 18 19 20 So far we have assumed that all variables were nonnegative (0 or positive). Now we want to think about them “possibly” being negative. Remember though we do not know what the variable represents so it could be positive or it could be negative. This shows that the number being squared could be 3 or it could be -3 and still get the same 9 21 2 So lets look at x we do not know whether x represents a positive or negative number. So when we bring it out of the radical we are not sure whether it should be positive or negative. If we are assuming that we do not know anything about the numbers being nonnegative then whenever a value comes out of the radical we have to put it inside absolute value bars. 22 23 24 25 26 27 28 29 30 31 Simplify Radicals This needs to be done on every single problem before you write an answer. The general accepted practice is to never write 2/4 as an answer we simplify to ½, the same applies here. If a radical can be written in a simpler form then we want to do so. When changing to a simpler form you should not impact the actual values represented. When I change 2/4 to ½ my calculator for both gives 0.5 this tells me that the simplification to ½ was right. We can use the same argument for checking our simplified radicals. When you simplify things you change the way it looks not the amount that it represents. If checking yourself on questions with variables you can still use your calculator just make sure that the values you choose for your initial problem are the same values that you use for your check. If you let x=2 in the initial problem you better let x=2 in your simplified version too. 12.2 Simplifying Radicals When simplifying radicals it is best to deal with the numerical parts separate. We will conduct what is called the PRIME FACTORIZATION of the radicand. -That means we will break the radicand down to the factors that multiply together to give the radicand. We will then group them based on the INDEX. 12 72 3 48 3 120 Now we can do the same thing with variables under the radical. You will want to re-write the variables that have higher exponents in multiples that relate directly to the INDEX For example… 5 2 x 3 y8 Now put it all together inside one problem Simplifying Fractions under a radical Break it up into two separate radicals. Numerator and denominator Then simplify each radical as far as possible. 5 9 25 81 • When you do this and a radical is left in the denominator you need to do what is called “Rationalizing the denominator”. To do this we multiply by a convenient value of 1. (that value will always be whatever the denominator is) • After you multiply you may have to reduce the radicals again. 25 72 • So far we have assumed that all variables were nonnegative (0 or positive). Now we want to think about them “possibly” being negative. • Remember though we do not know what the variable represents so it could be positive or it could be negative. • This shows that the number being squared could be 3 or it could be 3 and still get the same 9 2 do not know whether x • So lets look at we represents a positive or negative number. So when we bring it out of the radical we are not sure whether it should be positive or negative. • If we are assuming that we do not know anything about the numbers being nonnegative then whenever a value comes out of the radical we have to put it inside absolute value bars. x Rational Exponents 54 Rational Exponents To define what is meant by a rational exponent or, equivalently, a fractional exponent such as a1/3, we need to use radicals. To give meaning to the symbol a1/n in a way that is consistent with the Laws of Exponents, we would have to have (a1/n)n = a(1/n)n = a1 = a So by the definition of nth root, 55 Rational Exponents In general, we define rational exponents as follows. 56 Example 4 – Using the Laws of Exponents with Rational Exponents (a) a1/3a7/3 = a8/3 (b) Law 1: aman = am +n = a2/5 + 7/5 – 3/5 Law 1, Law 2: = a6/5 (c) (2a3b4)3/2 = 23/2(a3)3/2(b4)3/2 =( =2 )3a3(3/2)b4(3/2) Law 4: (abc)n = anbncn Law 3: (am)n = amn a9/2b6 57 Example 4 – Using the Laws of Exponents with Rational Exponents (d) cont’d Laws 5, 4, and 7 Law 3 Laws 1 and 2 58 Rationalizing the Denominator; Standard Form 59 Rationalizing the Denominator; Standard Form It is often useful to eliminate the radical in a denominator by multiplying both numerator and denominator by an appropriate expression. This procedure is called rationalizing the denominator. If the denominator is of the form , we multiply numerator and denominator by . In doing this we multiply the given quantity by 1, so we do not change its value. For instance, 60 Rationalizing the Denominator; Standard Form Note that the denominator in the last fraction contains no radical. In general, if the denominator is of the form with m < n, then multiplying the numerator and denominator by will rationalize the denominator, because (for a > 0) A fractional expression whose denominator contains no radicals is said to be in standard form. 61 Example 6 – Rationalizing Denominators Rationalize the denominator in each fraction. (a) (b) (c) 62 Example 6 – Solution (a) Multiply by (b) Multiply by 63 Example 6 – Solution (c) cont’d Property 2: 64 Rational Exponents 65 A square root is the number in which we multiply by itself 2 times to get the number under the root symbol. The number under the root symbol is called the Radicand. this is a symbol that represents the number that we multiply by itself 2 times to get 64. What is that number? 64 66 Likewise this is the symbol for the number that we multiply by itself two times to get 15. 15 out in front of the root When there is not a small number symbol then it is an understood 2. Which we call the square root. The number that is out front as part of the symbol is called the INDEX. 67 15 2 15 square root 3 8 cube root 4 81 fourth root 5 32 fifth root 68 The bottom of page 829 has a list of the most commonly used perfect squares, cubes, and 4th roots. 69 We cannot take the square root (or any even root) of a negative number. That is because you can never take two numbers (that are actually the same number) and multiply them together and get a negative number because when you multiply the same two numbers together then they are either both positive or they are both negative and in each instance the product would be positive. So there is no way to multiply and get -16 if the two factors have to be identical. 16 70 However, you can take the cube root (or any odd root) of a negative number because three negative numbers multiplied together will result in a negative number. 3 8 71 Sometimes there will be variables that are placed underneath the radical symbol. The general practice is to simplify the radicals, therefore we will try to reduce the exponents under a radical by drawing a connection between the INDEX of the radical and the exponent of the RADICANDS. 2 2 4 16a b 72 Simplify 2 81a 4b8 3 3 8x y 4 81a 4b8 9 73 Radicals can be written using exponents If x is a real number and n is a positive integer greater than 1, then 1n x x n 1 74 Examples to Simplify 12 9 271 3 49 12 (49) 12 12 16 25 75 Sometimes it is easier to write expressions using rational exponents and use properties of exponents (usually power to power) to simplify. 3 8 x3 y 9 4 81a 4b8 3 6 x y 12 76 The rational exponent theorem allows us to pull apart rational exponents (reverse of power to a power rule). For example. can be re written as (8) 13 2 23 8 32 25 9 2 3 77 Simplify the following x1 2 x1 4 x 35 56 z3 4 z2 3 ( x1 3 y 3 )6 x 4 y10 78 Adding / Subtracting Radical Expressions Multiplication / Division of Radical Expressions 79 Two radicals are said to be “like radicals” or “like terms” if they have the same index and the same radicand. Once this is true you can add/subtract the coefficients out front of the radical. 5 11 3 2 2 11 80 To recognize like radicands you may have to simplify one or more radicands. 3 2 4 8 2 18 81 82 Again you want to make sure that both expressions are simplified and rationalized. Example 2 83 84 85 Multiplying Radicals When multiplying radicals. The numbers on the outside get multiplied together. Then the numbers on the insides get multiplied together. At the end if the numbers started on the outside then their product stays outside, and if they started on the inside the product stays on the inside. You usually have to simplify the radicands when done. 86 Multiply the following 2 5 3 12 2 15 4 5 87 Sometimes you might have to foil or distribute the terms 3 12 8 2 5 4 3 5 2 2 4 6 5 14 88 89 90 91 Dividing of Radicals When division occurs with just one radical then we will rationalize through the use of multiplying by a value of 1. This value of 1 comes from whatever the denominator is. If the denominator is a sum or difference of terms that include radicals then we will use what is called a CONJUGATE. Conjugates are 2 groups of binomials that have the exact same terms but the middle signs are opposite. 92 Examples of Conjugates (a b)(a b) (2 r )(2 r ) ( x 3)( x 3) (4 2)(4 2) The nice thing about conjugates is that when you multiply (FOIL) the “O and I” terms cancel each other out. Which when dividing by radicals will produce a process that eliminates all radicals in the denominator. 93 94 95 96 97