In the previous section, we added and subtracted polynomials by combining like terms. In this section, we extend that idea to radicals.

Transcription

1 4.2: Operations on Radicals and Rational Exponents In this section, we will move from operations on polynomials to operations on radical expressions, including adding, subtracting, multiplying and dividing radical expressions. In the previous section, we added and subtracted polynomials by combining like terms. In this section, we extend that idea to radicals. Combining like terms can take on many forms. For example, 2xx + xx = 5xx 2xx 2 + xx 2 = 5xx 2 2 apples + apples = 5 apples 2 + = (2 sevenths + sevenths = 5 sevenths) = 5 7 In each of these examples, the object we are adding does not change after we add it. That is, when we add 2xx to xx we get 5xx, not 5xx 2. (Recall that the exponents change when we are multiplying the base, but not when we are adding.) Similarly, when we add like terms with radicals, the radicand (the number inside the radical) stays the same, so we get = 5 7, not 5 7 or This makes sense, because 7 can also be written as 7 1 2, and when we add, the exponents do not change: = Adding and Subtracting Radicals Similar to like terms in polynomials, like terms in radicals have two things in common: the same index and the same radicand. This means that we can only add or subtract radical terms together if the root is the same (square roots to square roots, cube roots to cube roots, etc.), and if the number on the inside of the radical is the same as well. EXAMPLES Add or subtract. Assume all variables are positive. a See if you can figure out the answer before turning the page

2 ( ) STEP 1: bring together like terms. We can combine the two numbers that have 6 as the radicand and we can combine the two numbers that have 5 as the radicand. STEP 2: add the coefficients (the numbers in front of the roots. Remember that 6 is the same as 1 6. We are finished because 6 and 5 are not like terms. b. aa 5 7aa 5 (aa 7aa + 8aa) 5 4aa 5 + 8aa 5 STEP 1: bring together like terms. Since all three have the same root and radicand, we can combine all three. STEP 2: add all three coefficients. c STEP 1: bring together like terms. But it looks like we don t have any like terms Then why would this problem even be here? TRY SIMPLIFYING FIRST That s right If we cannot add or subtract as is, first try simplifying each radical, then look for like terms again. Let s break this up into parts 8 =? 2 =? 18 =? See if you remember how to simplify these before checking on the next page.

3 8 = 4 2 = = 16 2 = = 9 2 = 2 So that = Now, notice that all of these are like terms So we can combine them into a single expression, and = = 2 d Again, we begin by simplifying each term in the expression, as there are no like terms yet 48 = 16 = 4 27 = 9 = = = 2 25 = 2 5 = 10 So that, = Again, now note that we have like terms, so we can combine them into a single expression, and we have = = 5 e Simplifying each term gives us: 45 = 9 5 = 5 = = 4 5 = 4 5 = = 4 5 = 2 5 = 2 5 Then, = = 15 5 f

4 Simplifying each term gives us: = 27 = 4 8 So that, 81 = 4 24 = 4 2 = = 8 8 = 5 g " = 7 2 = 7 2 = 7 2 = 7 2 = 7" 2 6" 16 = 6" 8 So that, " 16 = 6" 2 2 = 7" 2 = 12" " 2 = 19" 2

5 Multiplying and Dividing Radicals Property: ( ) = " " In other words, when we multiply two radicals together with the same index, we multiply numbers on the outside of the radical together, and then numbers on the inside of the radical together. EXAMPLES a. 5 (2 7 ) x 2 x 5 7 = 6 5 Multiply numbers on the outside of the radical, ( x 2) together, and numbers on the inside (5 x 7) together. Since we cannot simplify the square root of 5, we stop. b Multiplying numbers on the outside of the radical gives us 18 ( x 6), and numbers on inside together gives us 8 ( 2 x 4) 18 8 Notice that we CAN simplify the radical in this case, 8 = 2. So, 18 8 = 18 2 = 6 c ( 2 4 2) = In this case, we are multiplying a monomial by a binomial, so we distribute 2 to each of the other terms = = Note that you can simplify 4 4 = 4 2 = 8

6 d. + 5 FIRST: = OUTER: = INNER: 5 = 5 In this example, we are multiplying a binomial by a binomial, so we can use "FOIL" or the distributive property. LAST: 5 = 15 So, initially, we get: We can simplify because =, then we have: x , and we have like terms ( + 5 = 2 ), so the answer is: x e Again, in this example, because we have a binomial times a binomial, we can use FIRST: = 2 49 OUTER: 7 4 = 4 21 INNER: 2 7 = 6 21 LAST: 4 = 12 9 Initially, we get We can simplify 2 49 = 2 7 = 14, and 12 9 = 12 = 6. "FOIL" or the distributive property. So, we have However, note that we do have like terms, because = are like terms as well, and gives us 50. So the answer is: f. 5 2 Recall when we are squaring something first write it in expanded form, so that: 5 2 = Once again, we have a binomial times a binomial, so we can FOIL this as in the other examples, and we get:

7 FIRST: 5 5 = 25 OUTER: 5 2 = 10 " INNER: 2 5 = 10 " LAST: 2 2 = 4 Initially, we have " 10 " + 4, "#$%"&'"() " "#: " + 4 Dividing Radicals: Definition: Recall that when we divide radicals, it is not considered to be in standard form if there are any radicals in the denominator. Therefore, the main point in dividing radical expressions is going to be to get rid of the radical in the denominator. This process of getting rid of radicals in the denominator is also called RATIONALIZING (we call it rationalizing because we want to turn the denominator into a rational number but that s a horse of another color ). So, back to more examples. We know already how to divide (or rationalize) a fraction when there is one term in the denominator. But, what if we have 2 terms in the denominator, as in the example below? Well, if we have an expression in the denominator, then we multiply both the numerator and the denominator by what we call the CONJUGATE. Conjugate: The conjugate of any binomial term is found by negating the second term in the binomial. The conjugate of a + b is a b. The conjugate of x y is x + y. Let s look at some examples: EXAMPLES: Rationalize (divide) the following completely: g. We first have to find the conjugate of the denominator. The conjugate of + 1 is 1. We must multiply both the numerator and the denominator of the fraction by the conjugate

8 Now, we distribute the conjugate into the We distribute the 2 into each term in the numerator and we FOIL the denominator the numerator and the denominator. = We can now simplify the denominator. Notice that we can rewrite 9 ", and that "# cancel out So, we are left with: Note that there is NO radical left in the denominator This is what we wanted However, note as well that we can simplify these fractions a little further, each term in the numerator and denominator share a factor of 2, so we can simplify the fractions as below: = = 1 o. Again, we first have to determine what the conjugate of is, as before, the conjugate is just the change in the middle sign, so the conjugate of " +. Therefore, we multiply the numerator and the denominator by +. Multiplying these together gives us: = " + " = " "

9 p. We begin by finding the conjugate of the denominator. The conjugate of " +. Multiplying the numerator and denominator by the conjugate give us: + = =