MA 1128: Lecture 16 3/29/11 Rational Equations Roots and Radicals Equations with Rational Expressions (Example) Especially with the addition of rational expressions, we need a common denominator. Since we will be working with equations, we have the option of multiplying both sides by the same thing. When solving equations involving rational expressions, most people like multiplying both sides of the equation by the least common denominator (LCD). In this case, x2 + 7x + 12 = (x +3)(x + 4) is the LCD. 3 5 12 x 9 2 x 3 x 4 x 7 x 12 3 5 12 x 9

( x 3)( x 4) ( x 3)( x 4) 2 ( x 3)( x 4) x 3 x4 x 7 x 12 Next Slide Example (cont.) 3 5 12 x 9 2 x 3 x 4 x 7 x 12 3 5 12 x 9 ( x 3)( x 4) ( x 3)( x 4) 2 ( x 3)( x 4) x 3 x4 x 7 x 12

3( x 4) 5( x 3) 12 x 9 3 x 12 5 x 15 12 x 9 4 x 18 9 x 2 One thing that you should check for in a problem like this is to make sure the solutions are valid. Here, x = 3 and x = 4 would make the denominators zero in the original equation. This would be bad, so if either of these came up as a solution, we would throw them out. Next Slide More Examples In the first example below, the least common denominator is 10. In the other example, the least common denominator is (x + 1)(x 2). x x 3 5 2 x x 3 10

10 5 2 x 2 ( x 3) 5 2 x 5 x 15 3x 15 x 5 2 1 x 1 x 2 2 1 ( x 1)( x 2) ( x 1)( x 2) x 1 x 2 2 ( x 2) 1( x 1) 2 x 4 x 1 x 5 Next Slide

Practice Problems Find the solution to each of the following equations. 1. 3/8 + 3/4 = x/5. 2. (x + 1)/(x+10) = (x 2)/(x + 4). 3. (4x 1)/(x2 + 5x 14) = 1/(x 2) 2/(x + 7). Answers: 1) x = 45/8 2) x = 8 3) x = 12/5 Next Slide Radicals and Roots We talked about exponents awhile ago. Remember that we use them to indicate repeated multiplication. For example, 3333 = 34 = 81. It is convenient to have a notation for the inverse operation. In this case, if we have 34 = 81, then well say the fourth root of 81 is 3. In mathematical symbols, (the little house is called a radical symbol)

4 81 3 Since (-3)4 = (-3)(-3)(-3)(-3) = 81, -3 is also a fourth root of 81. To avoid any confusion, well make it clear which fourth root we want. 4 81 3 4 81 3 Next Slide Roots and Radicals (cont.) Since 42 = 16, well say that 4 is a second root of 16. Second roots are so common, they have a special name, and are called square roots. We also dont write the little 2. 2

16 16 4 16 4 A third root is usually called a cube root. Example, what is 3 8? What cubed is 8? Well, 222 = 8, so 3 8 2 Next Slide Practice Problems Give the positive root only. 1. 2.

3. 4. 4 3 27 49 4 16 Answers: 1) 2; 2) 3; 3) 7; 4) 2. Next Slide Rational (or Fractional) Exponents It turns out that we can write roots with exponents instead of radicals. They maybe dont look as nice, but theyre much easier to work with. Essentially, a second root is equivalent to an exponent of 1/2. A third root is equivalent to an exponent of 1/3. A fourth root is equivalent to an exponent of 1/4. For example,

1 3 x x 3 5 1 5 7 7 All of the usual rules for exponents apply. Next Slide With Calculators This notation works with our calculators really well, and it gives us options on how we use the calculator. Your calculator should have a square root button. To compute the square root of 16, you would enter 16, square-root button, and get the answer 4. (On some calculators, you hit the square-root button first.) You should also have an exponent button on your calculator (it probably has y x, xy, or ^ on it). [[That last exponent below is a .5]]

1 2 16 16 16 16.5 2 Enter 16, exponent button, .5, =, and the answer should be 4. Since you may not have a fourth-root button on your calculator, you can 4 1 4 16 16 16.25 Enter 16, exponent button, .25, =, answer is 2. Next Slide Practice Problems ROUND TO 4 DECIMAL PLACES. Youll need to be very careful to round correctly. For example, 1.732050808 would round to 1.7321, since a 5 in the fifth decimal place rounds up.

1. 2. 3. 4 625 5 10 4 2 Answers: 1) 5; 2) 1.5849; 3) 1.1892 Next Slide More on Rational Exponents Youll often see regular exponents mixed with exponents from roots. Just remember that the regular exponents go on top, and the roots go on the bottom.

5 5 3 5 3 x x x 4 7 3 7 4 7.75 4.3035 3 3 Rounded to 4 decimal places. Next Slide

Practice Problems Round to 4 decimal places. 1. 2. 5 53 35 Answers: 1) 2.6265 2) 15.5885 Next Slide Multiplying and Dividing with Radicals Note that radicals (roots) are essentially the same as exponents, so they distribute over multiplication and division. (BUT NOT ADDITION AND SUBTRACTION!!!)

For example, 4 4 2 25 25 5 3 2 3 1 1 1 3 16 8 8 2 4 x 2 4 x 2 2 x Next Slide Practice Problems

Simplify as much as you can. 1. 2. 3. 9 16 9x4 4 3 32 Answers: 1) ; 2) 3x2; 3) . End