Exponents and Radicals in Algebra

This section covers:

We briefly talked about exponents in the Powers, Exponents, Radicals (Roots) and Scientific Notation section, but we need to go a little bit further in depth and talk about how to do algebra with them.  Note that we’ll see more radicals in the Solving Radical Equations and Inequalities section, and we’ll talk about Factoring with Exponents, and Exponential Functions in the Exponential Functions section. Remember that exponents, or “raising” a number to a power, are just the number of times that the number (called the base) is multiplied by itself. Radicals (which comes from the word “root” and means the same thing) means undoing the exponents, or finding out what numbers multiplied by themselves comes up with the number. So we remember that  \(\sqrt{25}=5\), since  \(5\times 5=25\). Note that we have to remember that when taking the square root (or any even root), we always take the positive value (just memorize this).

Introducing Exponents and Radicals (Roots) with Variables

But now that we’ve learned some algebra, we can do exponential problems with variables in them!  So we have  \(\sqrt{{{x}^{2}}}=x\)  (actually  \(\sqrt{{{x}^{2}}}=\left| x \right|\)  since x can be negative) since  \(x\,\times \,x={{x}^{2}}\).   We also learned that taking the square root of a number is the same as raising it to  \(\frac{1}{2}\), so  \({{x}^{\frac{1}{2}}}=\sqrt{x}\).   Also, remember that when we take the square root, there’s an invisible 2 in the radical, like this:  \(\sqrt[2]{x}\). Also note that what’s under the radical sign is called the radicand (x in the previous example), and for the nth root, the index is n (2, in the previous example, since it’s a square root). With a negative exponent, there’s nothing to do with negative numbers!  You move the base from the numerator to the denominator (or denominator to numerator) and make it positive!  So if you have a base with a negative number that’s not a fraction, put 1 over it and make the exponent positive. And if the negative exponent is on the outside parentheses of a fraction, take the reciprocal of the fraction and make the exponent positive.  Some examples:   \(\displaystyle {{x}^{-2}}={{\left( \frac{1}{x} \right)}^{2}}\)  and   \(\displaystyle {{\left( \frac{y}{x} \right)}^{-4}}={{\left( \frac{x}{y} \right)}^{4}}\). Just a note that we’re only dealing with real numbers at this point; later we’ll learn about imaginary numbers, where we can (sort of) take the square root of a negative number.

Properties of Exponents and Radicals

So remember these basic rules:

Basic Exponential and Radical Rules

In algebra, we’ll need to know these and many other basic rules on how to handle exponents and roots when we work with them.  Here are the rules/properties with explanations and examples.  In the “proof” column, you’ll notice that we’re using many of the algebraic properties that we learned in the Types of Numbers and Algebraic Properties section, such as the Associate and Commutative properties. Unless otherwise indicated, assume numbers under radicals with even roots are positive, and numbers in denominators are nonzero.

Exponent Rules Radical Rules Rationalizing Fractions Denominators

I know this seems like a lot to know, but after a lot of practice, they become second nature.  You will have to learn the basic properties, but after that, the rest of it will fall in place!

Putting Exponents and Radicals in the Calculator

We can put exponents and radicals in the graphing calculator, using the carrot sign  (^) to raise a number to something else, the square root button to take the square root, or the MATH button to get the cube root or nth root.   Be careful though, because if there’s not a perfect square root, the calculator will give you a long decimal number that’s not the “exact value”.  The “exact value” would be the answer with the root sign in it! Here are some exponent and radical calculator examples:

Exponents and Radicals in the Graphing Calculator Rationalizing Radicals

Before we work example, let’s talk about rationalizing radical fractions.  In math, sometimes we have to worry about “proper grammar”.  When radicals, it’s improper grammar to have a root on the bottom in a fraction – in the denominator.  To fix this, we multiply by a fraction with the bottom radical(s) on both the top and bottom (so the fraction equals 1); this way the bottom radical disappears.  Neat trick! Here are some examples: Rationalizing Denominators Radicals Roots

Simplifying Exponential Expressions

There are five main things you’ll have to do to simplify exponents and radicals.  For the purpose of the examples below, we are assuming that variables in radicals are non-negative, and denominators are nonzero.

  • get rid of parentheses ().  Remember that when an exponential expression is raised to another exponent, you multiply exponents.  Also remember when you are multiplying numbers and variables and the whole thing is raised to an exponent, you can remove parentheses and “push through” the exponent.  Example:  
  • combine bases to combine exponents.  You should add exponents of common bases if you multiplying, and subtract exponents of common bases if you are dividing (you can subtract “up”, or subtract “down” to get the positive exponent as you’ll see).  Sometimes you have to match the bases first in order to combine exponents – see last example below.  Examples:  
  • get rid of negative exponents. To get rid of negative exponents, you can simply move a negative exponent in the denominator to the numerator and make it positive, or vice versa.  Examples:  
  • simplify any numbers (like = 2).   Also, remember to simplify radicals by taking out any factors of perfect squares (under a square root), cubes (under a cube root), and so on.  Example:  
  • combine any like terms.   If you’re adding or subtracting terms with the same numbers (coefficients) and/or variables, you can put these together.  Almost think of a radical expression (like ) like another variable.   Example:    .  Remember that, for the variables, we can divide the exponents inside by the root index – if it goes in exactly, we can take the variable to the outside; if there are any remainders, we have to leave the variables under the root sign.  For example, ,  since 5 divided by 3 is 1, with 2 left over (for the x), and 12 divided by 3 is 4 (for the y).

Now let’s put it altogether.  Here are some (difficult) examples.  Just remember that you have to be really, really careful doing these!

Simplify Exponents Negative Exponents Fractional Exponents

Here are even more examples.  Assume variables under radicals are non-negative.

More Examples of Simplifying Radical Expressions

If we don’t assume variables under the radicals are non-negative, we have to be careful with the signs and include absolute values for even radicals.  Here’s an example: Solving Exponent Equations with Even Radical and Negative

For all these examples, see how we’re doing the same steps over and over again – just with different problems?  If you don’t get them at first, don’t worry; just try to go over them again.  You’ll get it!  And don’t forget that there are many ways to arrive at the same answers!

Solving Exponential and Radical Equations

(We’ll see more of these types of problems here in the Solving Radical Equations and Inequalities section). Now that we know about exponents and roots with variables, we can solve equations that involve them. The trick is to get rid of the exponents, we need to take radicals of both sides, and to get rid of radicals, we need to raise both sides of the equation to that power. You have to be a little careful, especially with even exponents and roots (the “evil evens”), and also when the even exponents are on the top of a fractional exponent (this will become the root part when we solve). When we solve for variables with even exponents, we most likely will get multiple solutions, since when we square positive or negative numbers, we get positive numbers.  Also, all the answers we get may not work, since we can’t take the even roots of negative numbers. So it’s a good idea to always check our answers when we solve for roots (especially even roots)! Let’s first try some equations with odd exponents and roots, since these are a little more straightforward.  (Notice when we have fractional exponents, the radical is still odd when the numerator is odd). Solving Exponential Equations with Odd Exponents Now let’s solve equations with even roots.  Note that when we take the even root (like the square root) of both sides, we have to include the positive and the negative solutions of the roots.  (Notice when we have fractional exponents, the radical is still even when the numerator is even.) Again, when the original problem contains an even root sign, we need to check our answers to make sure we have end up with no negative numbers under the even root sign (no negative radicands). Also, if we have squared both sides (or raised both sides to an even exponent), we need to check our answers to see if they work. The solutions that don’t work when you put them back in the original equation are called extraneous solutions. Again, we’ll see more of these types of problems in the Solving Radical Equations and Inequalities section.

Solving Exponential Equations with Even Exponents

And here’s one more where we’re solving for one variable in terms of the other variables: Solving Exponential Equations Even Roots Variables Only

Note:  You can also check your answers using a graphing calculator by putting in what’s on the left of the = sign in “” and what’s to the right of the equal sign in “”.    You can then use the intersection feature to find the solution(s); the solution(s) will be what x is at that point. Here are those instructions again, using an example from above: Solving Exponential Equations with Graphing Calculator

Solving Simple Radical Inequalities

Note again that we’ll see more problems like these, including how to use sign charts with solving radical inequalities here in the Solving Radical Equations and Inequalities section. Just like we had to solve linear inequalities, we also have to learn how to solve inequalities that involve exponents and radicals (roots).  We’ll do this pretty much the same way, but again, we need to be careful with multiplying and dividing by anything negative, where we have to change the direction of the inequality sign. We also have to be careful that our answer still keeps what’s under an even radical to be positive, so we have to create another inequality and set what’s under the even radical to greater than or equal to 0, solve for x, and take the intersection of both solutions.  The reason we take the intersection of the two solutions is because both must work. With odd roots, we don’t have to worry – we just raise each side that power, and solve! Here are some examples:

Solving Radical Equations with Inequalities

Learn these rules, and practice, practice, practice!

For Practice: Use the Mathway widget below to try an Exponent problem. Click on Submit (the blue arrow to the right of the problem) to see the answer.

You can also type in your own problem, or click on the three dots in the upper right hand corner and click on “Examples” to drill down by topic.

If you click on Tap to view steps, or Click Here, you can register at Mathway for a free trial, and then upgrade to a paid subscription at any time (to get any type of math problem solved!).

On to Introduction to Multiplying Polynomials – you are ready!

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