# 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:

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.

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:

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:

# 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!

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

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:

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

And here’s one more where we’re solving for one variable in terms of the other variables:

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: