We will multiply, divide, and simplify radicals. The distance formula uses a radical. The Intermediate algebra
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1 We will multiply, divide, ad simplify radicals. The distace formula uses a radical. The Itermediate algera midpoit formula is just good fu. Class otes Simplifyig Radical Expressios ad the Distace ad Midpoit Formulas (sectio 10.) We will work with two importat rules for radicals. We will write them for square roots ut they work for ay root (cue root, fourth root, etc.). Product ad Quotiet Rules: For ay real umers a ad, we have a a. Likewise, a a. (This assumes that these radicals are real umers. I the case of the secod rule, the deomiators are o zero. Why?) It s always a good idea to ivestigate rules istead of elievig them lidly. Use the examples elow to verify the rules for yourself. expl 1: Puttig 4 i for a ad 5 i for, the product rule ecomes Simplify each side to verify the rule. This is just a matter of order of operatios. Do you multiply efore square rootig, or square root the multiply? The rule says it should ot matter. Multiply, ad the take the square root. Square root each, ad the multiply them. Were they equal? Does the rule hold true? expl : Puttig 64 i for a ad 16 i for, the quotiet rule ecomes side to verify the rule Simplify each Divide, ad the take the square root. Square root each, ad the divide them. 1 Were they equal? Does the rule hold true?
2 expl : Use the product rule to multiply the followig. Simplify if possile. a.) 5.) 10 x Rewrite as oe radical, ad the simplify what you ca. 4 4 c.) a 7a Rule works for ay idex: a a expl 4: Use the quotiet rule to divide the followig. Simplify if possile. a.) 6.) a a 7 6 Simplify radicals usig the ew rules: We will factor the radicad to fid perfect squares (or perfect th powers as i examples 6 through 8 elow). These ca e removed from the radical. Leftover its will stay uder the radical. expl 5: Simplify. Assume variales represet positive umers. 8 75x Look for perfect squares, cues, or th powers x x x x 5 is a perfect square. 7 is a perfect cue. 16 is a perfect 4 th power.
3 expl 6: Use the quotiet rule to divide the followig. Simplify if possile. 64x 5 10 x y y 5 7 Rule works for ay idex: a a expl 7: Simplify. Assume variales represet positive umers. 16x y Our ew rules allow us to look at the factors idividually. Fid a factor of 16 which is a perfect 4 th power. 4 x 4?? y y y?? 4 = 16 4 = = = 65 expl 8: Simplify. Assume variales represet positive umers. x 81y 1 Our ew rules allow us to look at the factors (o top ad ottom) idividually. Fid a factor of 81 which is a perfect rd power (cue). What do you do with the x o top?
4 Distace Formula: Below are two poits o a plae. If we wat to fid the distace etwee them, we ca draw i a right triagle (picture o right) ad use the Pythagorea Theorem. Notice i particular how the legs of the triagle are figured. The vertical leg turs out to e 8 1 or the differece of the y values. Why is the horizotal leg the differece of the x values? We will ow use two poits i geeral to derive the formula for the distace etwee them. We will agai use the Pythagorea Theorem. This is i this sectio ecause it will ivolve a radical. first poit (x 1, y 1 ) secod poit (x, y ) Pythagorea Theorem: c = a + What are a,, ad c? 4
5 Follow the steps to derive the distace formula. We start with the Pythagorea Theorem ad replace the variales with the pieces of our triagle. c = a + Put d i for the hypoteuse, ad the legs i for a ad. Square root oth sides to get d aloe. expl 9: Use the distace formula to fid the distace etwee the two poits (5, 4) ad (10, 6). Give a exact distace ad a three decimal place approximatio. d x x y 1 y1 Your poits are (x 1, y 1 ) ad (x, y ). Mid the order of operatios! Reduce the radical to get the exact distace. Use your calculator for the approximatio. 5
6 Midpoit Formula: This gives the poit i the exact middle of two other poits. (It is said to e the midpoit of the segmet coectig the two poits.) See the example elow. Ca you come up with a formula for the poits (x 1, y 1 ) ad (x, y )? Midpoit lies halfway etwee x values ad halfway etwee y values. Midpoit is the average of the two poits. What could the formula e? expl 10: Fid the midpoit of the lie segmet whose edpoits are (9, 5) ad (4, ). Also quickly plot the two poits with your midpoit to check yourself. 6
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