MTH 65 WS 3 ( ) Radical Expressions

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1 MTH 65 WS 3 ( ) Radical Expressions Name: The next thing we need to develop is some new ways of talking aout the expression 3 2 = 9 or, more generally, 2 = a. We understand that 9 is 3 squared and so we can say 9 is of 3. So, we have a way of explaining what 9 is with respect to 3, ut what if we want a way to explain what 3 is with respect to 9? What we say is that 3 is the square root of 9. Let s repeat that: 9 is of 3. 3 is of 9. More gernerally, if 2 = a we say that: a is of. is of a. So, what is the square root of 25? When I ask what is the square root of 49 what I m really asking is: Do we have a shorthand for asking what the square root of 36? You et! What we write is 36. In this notation, we call the racket thing around the numer the square root symol and the numer inside is called the radicand. So, what is: 1) 36 4) 100 7) ) 81 5) ) ) ) )

2 Pay close attention to the last two prolems. This shows us what? That is a + a + Okay, look at (6) again. Isn t the solution the same as if I d done 1 25? Of course it is! Why is this? Similarly, isn t the solution to (4) the same as 25 4? And, why is this? This rings us to our next 2 rules: The Product Rule for Square Roots a = a and a = a The Quotient Rule for Square Roots Let s use these to simplify the following: a = a and a = a 10) 45 12) ) ) 72 13) ) What is the phrase we use when simplifying as in (10) or (11)? vspace.25in So, we ve come to understand that the question eing asked when we say, what is the square root of 9 is really, what numer squared gives 9. What, then, is the answer to 1? Isn t this asking, what numer squared gives -1? Indeed we are asking that question and the answer is that there is no answer! In fact, whenever we have a negative inside the square root we cannot answer this question. What we say in this case is that there is no real solution. 2

3 16) 2 17) 9 18) 175 We learned earlier how to graph y = x 2, showing us that we can graph non-linear equations. If we re dealing with square roots, we can write a general question, x. That is, clearly (hopefully) we cannot simplify this since do not know the value of x. However, if we write y = x we can ask, what are solutions to this equation? We can say that the y value depends on the value we choose for x, and just like when we graphed y = x 2, we can set up a tale showing a set of solutions to this equation that we can use to get an idea of what the graph looks like. What is a solution to the equation y = x? If x = 0, what must y equal to have y = x? Use similar reasoning to fill out the following tale: x y = x Does it make sense to put a negative in the x column in the tale aove? Why or why not? Why didn t I put 2, 3 or 5 in the x column of the tale aove? Go ahead and graph the points from the tale aove and use these to sketch the equation y = x to the right of the tale. You will e expected to know how to graph this equation on a quiz or test at some point! Just a little f.y.i. Now, what do we do if there are variales involved? Note, again, we cannot do anything with x since we do not know the value for x. But what aout x 2? What is this asking? So x 2 = 3

4 This is an important fact that you will need to understand even more greatly in Math 95, ut for now just take note that taking the square root of x 2 gives you x ack. What is x 4 asking? So x 4 = What aout something like 4x 2? Well, here we can reak this up into two parts using the product rule for square roots: 4x2 = 4 x 2 and then we know how to address each of these! So, the final answer is 2 x or just 2x! One more quick one: How do we deal with x 3? Like in the last prolem, we re going to reak it up, ut how? Well, how aout using a similar technique as in the aove examples. We ll look for a perfect square to separate and then leave the rest under the square root. So x 3 = Okay, with these tools, let s do some more prolems: 19) 36x 2 21) 49y 5 23) 300 y 30 20) 12x 3 22) 50x 9 4x 4 24) 800x 12 10x 3 So we ve multiplied and divided square roots, what aout adding? We saw aove that a + a +, so can we simplify something like ? So, to review, we pulled out as much as we could from each term and then used a technique pretty much identical to comining like terms. 4

5 Here s some practice for you: 25) ) ) 4 2x + 8 2x 3 2x + 7x 29) 2 50x 2 18x 27) ) Excellent, now it s party time. How do you suppose we deal with something like 5( 3 + 6)? Normally, our order of operations tells us that we need to add inside the parenthesis first, however we cannot pull anything out of 6 or 3 and we cannot add them together since they don t share the same radicand. Thus, we treat the same way we would treat the prolem 5(x + 2). That is, we distriute through. For all of the next prolems, perform the indicated addition or multiplication, keeping in mind that we need to apply our distriution and foiling methods, where necessary, that were learned with inomials when the addition inside the parenthesis cannot e performed. 31) 5( 3 + 6) 36) ( )(5 3 6) 32) 6( 6 2) 37) ( x 11) 2 33) (7 + 2)(8 + 2) 38) (2 3x y) 2 34) ( 6 + 3)( ) 39) 7 32x 3 3x 50x 35) ( )( 11 6) 40) 9y 2 x 5 y + x 2 xy 5 5

6 The next wonderful thing that we get to learn is called Rationalizing the Denominator. What does this mean? So how can we accomplish this feat? Well, what happens if I multiply 2 y itself? Our multiplication rule aove tells us that 2 2 = 2 2 = 4 = 2 Real quick, efore getting ack to rationalizing the denominator, this is a good place to point out that if we also note that 2 2 = ( 2) 2 we end up with the relationship ( 2) 2 = 2 This is another example of something that can e stated more generally. That is ( a) 2 = a a = a a = a 2 = a Or, without the inetween steps: ( a) 2 = a This ties right in with the fact that a 2 = a is going to important in Math 95. Okay, ack to rationalizing the denominator. The important thing we need in order to rationalize the denominator is the fact that a a = a. So if I have a square root in the denominator i can do what to get rid of it? Great, let s try a few! 41) ) 2x 17 42) x ) 27x 2 12y 3 6

7 There s one other situation in which we will learn how to rationalize the denominator. When we have a sum in the denominator with one of the terms having a square root in it, we can use a technique called: on the top and ottom of the fraction, which will clear the square root. Say I want to rationalize the denominator of Then I need to multiply top and ottom y the conjugate of So what is the conjugate? In general, the conjugate of a + = a, the conjugate of a = a +, the conjugate of a + = a and the conjugate of a = a +. Essentially, we are just changing the sign etween the two terms. Why does this help us? Go ahead and FOIL out the following: 45) ( a )( a + ) The square root disappeared! That w why this helps us! Let s try rationalizing the denominators of the following: 46) )

8 48) ) 2x+4 2h x+2 h 49) ) 3 x+3 x Alright, it s everyodies favorite time! Story prolems! Police us the formula ν = 20L to estimate the speed of a car, ν in miles per hour, ased on the length, L, in feet, of its skid marks upon sudden raking on a dry asphalt road. Use the formula to solve the next Exercise. 52) A motorist is involved in an accident. A police officer measures the car s skid marks to e 45 feet long. Estimate the speed at which the motorist was traveling efore raking. If the posted speed limit is 35 miles per hour and the motorist tells the officer she was not speeding, should the officer elieve her? Explain. 8

9 Okay, so we ve answered the question, what squared equals 16, or some such numer, ut what if we wanted to answer the question, what cued equals 27? That is, what is the solution to x 3 = 27? Or, what to the fourth equals 16? (What s the solution to x 4 = 16?). We could continue for any power of x! What is the solution to x 8 = 256? And, so on! While with 3 2 = 9 we say that 9 is the square of 3 and that 3 is the square root of 9, with 4 3 = 64 we say that 64 is the cue of 4 and 4 is the cue root of 64. We can say 625 is the 5 to the fourth or we can say 5 is the fourth root of 625. The notation for this is similar to the square root: The cue root of 8 we write 3 8 which is 2 since 2 3 = 8. The fourth root of 81 we write 4 81 which is 3 since 3 4 = 81. In general we say that n a = means that n = a Note that n here is called the index. The index for square roots is 2 ut is usually omitted. There are rules for multiplying, dividing and adding other roots that are similar to the rules for square roots and for similar reasons. The important thing to rememer when dealing with these expressions is: Before getting into all these rules it is important to point out that while 1 has no real solution, 3 1 does have a solution, namely -1 since ( 1) 3 = 1. Why is this? So, what aout 4 1 or 5 1? The general idea here is that if the index n is even, there is no solution to n 1 ut if n is odd, there is a solution. This is not only true for n 1 ut for the nth root of any negative numer! Why is this? The Product and Quotient Rules for nth Roots are: n a n = n a and n a = n a n n a n = n a, 0 and n a = n a n, 0 9

10 An important thing to note here is that the indices must e the same. Please take a moment and rewrite this here. Similarly, for adding, for two roots to e like they must have the same index and radicand. With these in mind, let s do some examples. 53) ) ) ) ) 4 10, ) ) ) 3 16x 4 57) ) x 3 48x x 4 10

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