Lesson 7-5: Solving Radical Equations

Transcription

1 Today we re going to pretend we re Ethan Hunt. You remember Ethan Hunt don t you? He s the Mission Impossible guy! His job is to go in, isolate the bad guy and then eliminate him. Isolate and eliminate mission impossible that s what we re doing today. Eponents the bad guy You are getting pretty good at solving equations. You know the drill: 1. What s with the?. How is it combined with the?. How do I undo it? 4. If I do something to one side, I have to do it to the other side. If you think about it, these steps are focused on getting the all by itself: isolating it and eliminating the stuff with it. This is fairly easy to do with a linear equation like 4 4. You can even solve an equation like 4 4 by using the square root to eliminate the squared part. But what if the variable has a rational eponent like this: 4 14? How in the world do we solve that? Undoing / eliminating a rational eponent If you had the equation 4, how would you isolate the? What is with it? There is a / with the. How is it combined? With multiplication. How do we eliminate it? Well, one way is to multiply both sides by /. Why? Because / times / equals 1. If we multiply both sides by the reciprocal, we eliminate the fraction with the variable. How does that apply with rational eponents? Let s work with our equation 4 14 : and now we re kinda stuck. There is a / with the but it is an eponent. We want to eliminate it but how? How do undo that eponent? Page 1 of 6

2 Well, we want to have an = type of equation right? Let me ask you this: what is the eponent in our desired equation? It is 1; there is always the hidden eponent of 1! So, how do we turn that / into a 1? Easy! Multiply it by / : / / = 1 Okay so we multiply the eponent of the by ; that means we re squaring the left side of the equation. If we do something to one side, we need to do it to the other side right? So we need to square the other side too! Let s continue solving our equation: 9 We want to get rid of the rational eponent with the multiply the eponents by the reciprocal take care of the remaining rational eponent by splitting it up and simplify Now we have our answer double check it. Take the 16 and plug it back into our original equation to see if it works: Yup, it checks out! We re good to go! Undoing / eliminating a radical If you had the equation 1 6, how would you isolate the? This time it is inside a radical root. How do we undo that? It is easy if you remember that 1 is the same thing as 1 1. How do I eliminate the fractional eponent? I multiply by the reciprocal in this case. So if I see a radical equation, all I need to do is raise both sides to the same power as the root. Here is how it would work: 1 6 Theroot is 1 6 Raiseboth sidestotherd power to get rid of the radical 1 16 Now just simplify 15 Page of 6

3 Let s practice! Following are nine eamples showing the solution process with different setups. At the end I will show you two different applications that use this type of process: the distance formula and geometric mean Get rid of the eponent by multiplying the eponents by its reciprocal 16...now just simplify 8 : Don't forget there is a hidden in the elbow of the radical... 5 Get the radical by itself by adding 5 to both sides...now get rid of the radical 8...by raising both sides 64 : nd to the power. 0 0 Get the radical by itself......by adding to both sides...now get rid of the radical 8 rd...by raising both sides to the power : Get the radical by itself by subt 7 from both sides...get rid of radical th 1...by raising both sides to the 4 power and simplify : Page of 6

4 Get rid of the eponent by multiplying by the reciprocal break apart the rational eponent 1 :9 9...and simplify Get the radical by itself by subt 1 both sides...net get rid of radical by squaring both sides and simplify : Separate the radicals by moving the negative radical to the other 5...get rid of the cubed root 5...be careful on the left side! Cube all parts! 85...and simplify : side Page 4 of 6

5 Okay, here is one with a little twist. Before you try to solve it, look at and think about the problem. Will the square root of a number ever be negative? In other words, is there any number which times itself that is negative? Err, nope. What is my point? Well, let s walk through the solution process and see what happens : and it doesn' t work! Why not??? So it didn t work let s think about this. Even before we worked the solution, we d decided that you can never get a negative number out of a square root. We worked the solution process correctly but still came up with a wrong answer. What did we do wrong? Nothing we did nothing wrong. This is what is called an etraneous solution. It is a solution obtained from correctly following the solution process that does not work. What is the point? Always, always, always check your answers! Get the radical by itself by adding 1 to both sides simplify a get rid of the radical by squaring both sides FOIL out the right side... nd use quadratic formula to solve for b b 4ac ,, a 1 Check both answers! : it works : it works The distance formula The distance between two points is given by the formula d y y where the two points are,, y and y Page 5 of 6

6 Try it out: find the distance between,1 and 5, First, identify 1,, y 1, and y : d y y y y The key with the distance formula is plugging the right numbers in the right place. I always write out 1 =, =, y 1 =, and y = to help me keep it straight. I suggest you do the same. If I need to, you need to. Geometric mean If you recall, when we talk about the mean of a set of numbers, we add them and divide by the number of them. This is an average of the numbers; another name for it is the arithmetic mean. If you take the arithmetic mean (or average) of 1,, 5, 7, 9, 11 and 100, it will be significantly affected (skewed) by the unusually big number: the arithmetic mean is The 100 value pulls the mean higher; if you take the 100 out, the arithmetic mean is 6! There is another type of mean that is very useful for averaging numbers when the list includes numbers that are much bigger than the others. It is called the geometric mean and is given by the formula n a1 a a n. The geometric mean of our eample above is As you can see it softens the effect of an etreme number, moving it much closer to the rest of the numbers. Geometric mean is frequently used in financial situations to soften the effect of unusually large profits, epenses or value. Now, what if you knew the geometric mean of 8 and another number is 4. What is the other number? The geometric mean 4 will be found by 8 : 4 8 The geometric mean is the square root of 8 times Get rid of the radicals...square both sides and simplify 48 6 The other number is 6! 8 Page 6 of 6