Idiot's Guides: Algebra I (2015)
Part V. Radical, Quadratic, and Rational Functions
The last portion of one of my detective stories is always the most interesting part for me. The heroes have finished their investigation, overcome the obstacles, debunked the false leads, and now they’re ready to unravel the mystery for us. They explain all the twists and turns, show us how this amazingly complicated problem breaks down to an application of things we already knew, and leave us thinking, “Of course! That makes so much sense!”
This part of the book is our chance to bring all the algebra you already know to explaining some problems that may seem mysterious at first. The graphs of square root functions, quadratic functions, and rational functions look very different from the lines we graphed earlier, but we’ll be able to use many of the same strategies for graphing them. Solving equations related to these functions will also break down to simpler problems if we use techniques we learned earlier. It’s time to tie all the pieces together.
Chapter 14. Radical Functions
In This Chapter
· Simplifying radicals
· Rationalizing denominators
· Doing arithmetic with radicals
· Solving radical equations
In arithmetic, we first learn to work with numbers and the four basic operations, but soon we introduce exponents, or powers, into the work, to shorten repeated multiplication. With exponents come roots, a way to undo a power. The symbol for a root, the radical sign, gives its name to the operation. The square root of 2, written , is often read as “radical two.”
Roots of numbers are numbers, and you’ve already learned a bit about how to simplify them. When variables start to appear under the radical sign, simplifying becomes a little more difficult but even more important. When variable expressions under radicals appear in equations, the tactics you’ve used for linear equations are no longer enough. In this chapter, we’ll look at the process of simplifying radicals that contain variables, and we’ll develop some new tools for solving radical equations.
Exponents tell how many times to use a number in a multiplication. If you write 35, you’re saying you want to multiply five 3s. 35 = 3·3·3·3·3 = 243. You can raise a number to any power you need, but the power you’ll most commonly see is the second power, or square. Multiplying 9.9 can be written as 92. Formally, that’s read as “9 to the second power,” but casually we say “9 squared.”
THINK ABOUT IT
Raising a number to the second power came to be known as squaring because finding the area of a square requires multiplying the length of a side by itself. You raise the length of a side to the second power, or square the side, to find the area of a square. To find the volume of a cube, you raise the length of an edge of the cube to the third power, which is why the third power is called the cube and raising a number to the third power is cubing.
Roots are the inverses, or opposites, of powers, and just as you can raise a number to any power, you can take any root. You know 35 = 243, so the fifth root of 243 is 5. The fifth root tells you what number you would raise to the fifth power to produce 243. The symbol for the fifth root of 243 is . The small number 5 in the crook of the radical sign is called the index. It tells what power is being undone. If you don’t see an index, it’s 2. The square root, the inverse of the second power is the most common, so if you don’t see an index, you can assume it’s a square root. The number under the radical sign is called the radicand.
The index is the small number that appears in the crook of the radical sign and tells what power is being undone. If no index appears, the radical indicates a square root.
Working with radicals is easier if they are in simplest radical form. Getting to that form takes a few steps. The first step is to get the smallest possible number under the radical. Use the rule that . Think about numbers whose square root you know. and . If you multiply , you’re multiplying 2.3, which is 6, and 6 is , which is .
The method of simplifying radicals works for all indices, not just square roots, but to simplify a third root, for example, you must look for numbers that are perfect cubes, not perfect squares. If you were simplifying a fifth root, you’d look for factors that were integers raised to the fifth power.
To get to the smallest number under the radical, look for factors of the radicand whose square root you know. To simplify , list the possible factor pairs for 48.
48 = 1·48
There are two numbers on that list that are perfect squares, 4 and 16. Use the larger one to write as and then use the rule to say . Because you know that , you can say . The simplest form of is .
To put a radical in simplest form, find the largest perfect square that is a factor of the radicand. Rewrite as a product of two radicals and take the square root of the perfect square.
To leave the smallest possible number under the radical, look for the largest perfect square factor. The fastest way to find the largest perfect square, however, is to start with a small factor and see if the factor-pair partner is a perfect square. If you need to simplify , don’t grope for perfect square factors. Think 432 ÷ 2 = 216, not a perfect square, but 432 ÷ 3 = 144 and 144 is the largest perfect square factor.
When variables appear in the radicand, it becomes a little more challenging to simplify, because you don’t know what the variable represents. can’t be simplified any further. The number that x represents might be a perfect square or might be a number with a perfect square factor, but you just don’t know that. There are steps you can take, however.
If the variable has a numeric coefficient, you can simplify that. and . If the variable under the radical is raised to a power, you may be able to work with that to make the radicand smaller.
Roots undo powers, so taking the square root of a square should get you back where you started, and it does, with one caution. When you take a square root, you probably think of a positive number. In fact, it is agreed a radical in print will mean the principal, or positive, square root. will mean 3, but we know that (−3)2 is also equal to 9. By agreement, if we want to talk about the negative square root, we put a negative sign in front of the radical. . If we want both the positive and the negative square roots, we write .
When it comes time to take the square root of x2, as in , the quick response is x. The problem is, we don’t know what number x stands for. Suppose x were −5. (−5)2 = 25 and but x is not 5. It’s -5. In that case (and many others), saying wouldn’t be quite true. might equal x, or it might equal the opposite of x, whichever is the positive number. To guard against this problem, we will say .
THINK ABOUT IT
If you have a guarantee that x is positive, you can drop the absolute value signs. Sometimes that guarantee will be in the directions (Assume all variables are positive). Sometimes the situation may tell you negative numbers aren’t reasonable, for example, when the variables represent lengths of sides.
Now you can apply a strategy for simplifying radicals to radicands that are variable terms. To simplify , factor the radicand, using perfect squares as factors whenever you can.
Apply the rule.
Rearrange a bit, and apply the product rule in the other direction to get back to just one radical.
There is no rule for the root of a sum except to find the sum and take its square root. does not equal . Putting numbers in place of x and y will show you that. but . Clearly, they’re not equivalent.
Put each radical in simplest form.
Making the radicand as small as possible is one step in simplifying a radical expression, and the one used most often. If the expression under the radical is a fraction or if the radical expression is in the denominator of a fraction, however, you have more work to do.
The basic rule used to simplify radicals is . When dealing with fractions and radicals at the same time, we’ll count on a related rule: . If we need to take the square root of , this rule states that . It also means that we can go the other way. .
Sometimes, however, things are not that tidy. If you need to take the square root of , you can say and you know that the square root of 1 is 1, but the square root of 2 is an irrational number. You don’t want a non-terminating, non-repeating decimal in the denominator, and if you start rounding, you won’t have an exact answer. So it seems like you’re stuck with .
One of the requirements for simplest radical form is that there are no radicals in the denominator. Radicals in the numerator are okay, but not the denominator. How can you get to simplest radical form? By using a familiar method to change the way a fraction looks without changing its value: multiply by 1.
THINK ABOUT IT
Why are we willing to have radicals in the numerator but not in the denominator? The answer has to do with being able to have a sense of numbers without having to carry a calculator around constantly. If I write , you can think “ is about 1.7, so is about 0.85.” It’s not exact, but you have a sense of the number. If, on the other hand, I write , even the estimated value of as 1.7 doesn’t help all that much, because you probably can’t divide 2 by 1.7 in your head. It’s more than 1, less than 2, but that’s about the best we can do. But if we rationalize the denominator, .
To create a common denominator, multiply the numerator and denominator of the fraction by the same number. If you multiply , you create a very different looking fraction, , that has the same value as . The reason that the value doesn’t change is that the multiplier, , is equal to 1. Let’s use a similar multiplier to get rid of the radical in the denominator of . Multiply by , which is equal to 1.
Clearing the radicals from the denominator of a fraction (even if it means putting radicals in the numerator) is called rationalizing the denominator. If the denominator of a fraction contains a single radical term, you can rationalize the denominator by multiplying the numerator and denominator by the radical of the denominator. It’s not necessary to include any multiplier outside the radical. You can rationalize the denominator of by multiplying by . You don’t need to use the 5.
Rationalizing the denominator is the process of changing the appearance, but not the value, of a quotient so that no radicals remain in the denominator.
It is possible for the denominator of the fraction to be a sum or difference of two terms, one or both of which is a radical. In that situation, the strategy you just used won’t work. If you multiply by , the radical in the denominator moves, but it doesn’t disappear.
You haven’t rationalized the denominator. If fact, things have gotten a little worse. When the denominator is a sum or difference, you will need to use the conjugate of the denominator. The conjugate is the same two terms with the opposite sign connecting them. The conjugate of is and the conjugate of is .
The conjugate of an expression that is the sum of two terms is an expression that is the difference of those two terms. If a − b is the conjugate of a + b, then a + b is the conjugate of a − b, and the product (a − b)(a + b) = a2 − b 2.
You need to multiply by . To do that multiplication, you need the distributive property and the FOIL rule covered in Chapter 2.
Once you start canceling, it can be hard to stop. If you have , it’s tempting to think you could cancel the 15 with the 5, but that wouldn’t be correct. You can cancel a factor from the numerator with a factor from the denominator, but the 15 is a term, not a factor. It’s connected to by addition. Remember the fraction bar acts like a grouping symbol. You need to divide 15 by 5 and by 5. , not . Never cancel terms.
Rationalize denominators and put each expression in simplest radical form.
Operations with Radicals
You’ve already done some multiplication with variables, and there really isn’t any division. If you have a division problem involving radicals, you just write it as a fraction and put it in simplest form, being sure to rationalize the denominator.
When it comes to adding and subtracting, radicals act a lot like variables. You can only add or subtract like radicals, just as you can only combine like variable terms. You perform the operations by adding or subtracting the coefficients in front of the radicals. just as 6x + 9x = 15x.
If you’re trying to combine variable terms and they don’t have the same variable, there’s nothing you can do. 5x + 3y just can’t be simplified. When you’re working with radicals, however, you don’t want to jump to conclusions. At first glance, it may seem that is impossible to simplify, but each of those radicals can be simplified.
In their original form, the radicals appeared to be unlike, but once they’re simplified, you can see that both are multiples of , and can be combined.
Simplify each expression by performing the indicated operations. Simplify radicals when necessary.
Solving Radical Equations
When you encounter radicals in equations, you may need to add a new technique to your toolbox for solving equations. If the radical is just the square root of a constant, you can treat it like any other number. When the radicand includes a variable, on the other hand, you need a new way to get to a solution.
Equations with One Radical
If you’re facing a linear equation like 6x − 7 = 11, you can use inverse operations to isolate x.
When you meet a radical equation like , you need to free the 6x − 7 from that radical, so that you can get to work isolating x. To do that, use an inverse operation. The inverse of taking a square root is squaring, so begin by squaring both sides.
Squaring a radical makes the radical “lift” or disappear, and leaves you with just the radicand. Squaring the 11 on the other side gives us 121. The important part is that you now have a linear equation that you can solve as usual.
THINK ABOUT IT
Why does squaring both sides make the radical disappear? Squaring and taking the square root are opposite or inverse operations. Each will undo the work of the other. If you take the square root of a number () and then you square that answer (72 = 49) you get right back to the number you started with. When you square a square root, you get the radicand. The radical lifts.
When you solve a radical equation, the first important additional step is squaring both sides to remove the radical. There are a couple of other concerns that are important, too. One is to be certain to check the solution. The squaring of both sides that’s critical to the whole process can also introduce what are called extraneous solutions. They’re not mistakes. You’ve done your algebra correctly, but they don’t work. Sometimes you get a perfectly normal solution; sometimes you get an extraneous solution. Later, when we look at equations that have more than one solution, you’ll see that you might get two usable solutions, two extraneous solutions, or one of each. Whenever a radical is involved, it’s crucial to check the solution.
An extraneous solution is a solution produced by correct algebraic procedures that does not satisfy the equation.
So let’s check the solution of x = for the equation . Replace x in the original equation with , and simplify.
The solution checks.
The other thing to keep in mind when you’re solving radical equations is the radical must be isolated on one side before you square both sides. Suppose that, instead of , you needed to solve . Only the 6x is under the radical. If you tried to square both sides right now, you would have a problem, a couple of problems actually.
Problem 1 is that to square , you’re going to have to use the FOIL rule. and that’s a lot of work. Take a look at what happens if you put in that work.
Not only have you not eliminated the radical, you’ve produced something even more difficult to solve than the equation you started with. This is not a path you want to take.
Instead, take a moment to isolate the radical. In this equation, add 7 to both sides.
Now you can square both sides and the radical will lift.
That’s easier, right? Check the solution before moving on. .
To solve an equation with one radical:
1. Isolate the radical.
2. Square both sides.
3. Solve the resulting equation.
4. Check your solution.
Equations with Two Radicals
Solving radical equations depends on eliminating the radicals, so it’s natural that an equation that has more than one radical would look challenging. In fact, they’ll yield to the method we’ve outlined. It just requires patience and persistence.
If you want to solve , focus on isolating the first radical. That will give you , and squaring both sides will lift both radicals.
The solution of x = 5 does check, and that wasn’t really any more work than the equations with one radical. The key to that simplicity was the zero in the original equation. Let’s look at one that’s not quite so simple.
To solve the equation , start the same way, by isolating one radical. Then square both sides.
This is going to require the FOIL rule.
Now isolate the other radical. Remember: patience and persistence.
Square both sides yet again, and solve.
Extraneous solutions often occur when there are variables on both sides of the equation. Here’s an example of a radical equation that has two solutions, but one is extraneous. The solution below will be explained in a later chapter, but the important piece is that there are two solutions.
The solution of x = 10 checks: . The solution of x = 5 is extraneous. but 5 − 7, not 2.
Solve each equation. Be certain to check for extraneous solutions.
The Least You Need to Know
· Simplify radical expressions by finding the largest perfect square factor of the radicand, and apply the rule.
· Rationalize a denominator that is a single term by multiplying the numerator and denominator by the radical that appears in the denominator.
· The conjugate of an expression of the form a + b is an expression of the form a − b.
· Rationalize a two-term denominator by multiplying the numerator and denominator by the conjugate of the denominator.
· Solve radical equations by isolating the radical and squaring both sides.
· Always check for extraneous solutions when solving radical equations.