We briefly talked about exponents in the **Powers, Exponents, Radicals (Roots) and Scientific Notation** section, but now we’ll work with them using algebra. Note that we’ll see more radicals in the **Solving Radical Equations and Inequalities **section, and we’ll talk about **Factoring and Solving with Exponents **here in the **Advanced Factoring** section. Exponential Functions are discussed in the **Exponential Functions** section. Also, **imaginary** (non-real) roots are discussed in the **Imaginary (Non-Real) and Complex Numbers **section.

## Introducing Exponents and Radicals (Roots) with Variables

Here are some basics of Exponents and Radicals, as we start to use them in Algebra:

- An
**exponent**, or “raising” a number to a power, is just the number of times that a**base**is multiplied by itself. In this example, the exponent is**3**and the base is**5**: $ {{5}^{3}}=5\times 5\times 5=125$. **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. For example, $ \sqrt[3]{{125}}=5$*,*since $ 5\times 5\times5=125$*.*- When we take the square root, there’s an invisible
**2**in the radical, like this: $ \sqrt[2]{x}$). - When taking the square root (or any even root), we always take the
**positive value**(just memorize this). For example, $ \sqrt{4}=2$.**But when solving for an even root in an equation, we have to take plus and minus; for example:**$ {{x}^{2}}=4;\,\,x=\pm 2$. - $ \sqrt{{{x}^{2}}}=\left| x \right|$ since $ x$ can be negative; try it: $ \sqrt{{{{{\left( {-3} \right)}}^{2}}}}=\sqrt{9}=3=\left| {-3} \right|$. More generally, $ \sqrt[n]{{{{x}^{n}}}}=\left| x \right|$, if $ n$ is even. Note that $ \left| {} \right|$ signifies the
**Absolute Value**of a number, which is the positive value only. - Taking a root of a number is the same as raising it to $ \displaystyle \frac{1}{{\text{that root}}}$; for example, $ \displaystyle {{x}^{\frac{1}{2}}}=\sqrt{x}$.
- A
**negative exponent**has nothing to do with negative numbers as we know them! Move the base from the numerator to the denominator (or denominator to numerator) and make the exponent positive! If the negative exponent is on the outside of the parentheses of a fraction, take the**reciprocal**of the fraction (base) 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}}$. - The two basic ways to write radicals (roots) are via a
**radical expression**(such as $ \sqrt[3]{x}$), and a**Rational expression**(such as $ {{x}^{{\frac{1}{3}}}}$; “rational” means fractional). I remember this since the $ \sqrt{{}}$ is a radical sign, and rational sounds like fractional. As another example, the rational expression $ \displaystyle {{x}^{{\frac{3}{5}}}}$ can be written in radical form as $ \displaystyle \sqrt[5]{{{{x}^{3}}}}$ or $ \displaystyle {{\left( {\sqrt[5]{x}} \right)}^{3}}$; note that the exponent can be either inside or outside the radical. - What’s under the radical sign is called the
**radicand**, and the**index**is the actual root. In the example $ \sqrt{x}$, $ x$ is the radicand and (“invisible”)**2**is the index.

Note that in these discussions, 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

To summarize, here are some **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!

## 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 $ n$th 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. **You need to know your calculator**!

Here are some exponent and radical calculator examples (**TI 83/84 Graphing Calculator**):

Using a **TI30 Scientific Calculator**, here are the steps:

## Rationalizing Radicals

In math, sometimes we have to worry about “proper grammar”. With radicals, it’s improper grammar to have a root in the denominator of a fraction, and thus we have to “**rationalize**” it. To rationalize, multiply by a fraction by **1** such that the denominator “cancels” out the radical. If two terms are in the denominator, multiply the top and bottom by a **conjugate**. Here are some examples:

## Simplifying Exponential Expressions

There are some hints for simplifying 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****()**. When an exponential expression is raised to another exponent, multiply exponents. When an algebraic expression, for example, with coefficients and variables in parentheses, is raised to an exponent, remove parentheses and “push through” the exponent. Example: $ {{\left( {6{{x}^{3}}y} \right)}^{2}}=36{{x}^{6}}{{y}^{2}}$.**Combine bases to combine exponents**. Add exponents of common bases if you are multiplying, and subtract exponents of common bases if you are dividing (you can subtract “up”, or subtract “down”,**starting with the largest exponent**, to get the positive exponent). Sometimes you have to**match the bases first**in order to combine exponents. Examples: $ \displaystyle \begin{array}{c}{{a}^{2}}{{a}^{3}}={{a}^{5}};\,\,\,\displaystyle \frac{{{{a}^{5}}}}{{{{a}^{3}}}}={{a}^{{5-3}}}={{a}^{2}}\,\\\displaystyle \frac{{{{a}^{3}}}}{{{{a}^{5}}}}=\displaystyle \frac{1}{{{{a}^{{5-3}}}}}=\displaystyle \frac{1}{{{{a}^{2}}}}\,\,(\text{which is }{{a}^{{-2}}})\end{array}$, $ \displaystyle \frac{{{{9}^{3}}}}{{{{3}^{{-4}}}}}=\frac{{{{{\left( {{{3}^{2}}} \right)}}^{3}}}}{{{{3}^{{-4}}}}}=\frac{{{{3}^{6}}}}{{{{3}^{{-4}}}}}={{3}^{{6-\left( {-4} \right)}}}={{3}^{{10}}}$.**Get rid of negative exponents**. To get rid of negative exponents, simply move a negative exponent in the denominator to the numerator and make it positive, or vice versa. Examples: $ \displaystyle {{a}^{{-4}}}=\frac{1}{{{{a}^{4}}}};\,\,{{\left( {\frac{a}{b}} \right)}^{{-2}}}={{\left( {\frac{b}{a}} \right)}^{2}}=\frac{{{{b}^{2}}}}{{{{a}^{2}}}}$.**Simplify any numbers**(like $ \sqrt{4}=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: $ \sqrt{{50{{x}^{2}}}}=\sqrt{{25\cdot 2\cdot {{x}^{2}}}}=\sqrt{{25}}\cdot \sqrt{2}\cdot \sqrt{{{{x}^{2}}}}=5x\sqrt{2}$.**For exponents inside root signs, divide exponents by their root index, and if it goes in exactly, move to the outside with as the exponent. Even if it doesn’t go in exactly, move highest factor to outside, and leave remainders under the root sign. For example,**$ \sqrt[3]{{{{x}^{5}}{{y}^{{12}}}}}={{x}^{1}}{{y}^{4}}\sqrt[3]{{{{x}^{2}}}}=x{{y}^{4}}\sqrt[3]{{{{x}^{2}}}}$**,**since, for $ {{x}^{5}}$,**5**divided by**3**is**1**, with**2**left over, and for $ {{y}^{{12}}}$,**12**divided by**3**is**4**. You can also use**rational**(**fractional**) exponents to see this, which is sort of like turning improper fractions into mixed fractions: $ \sqrt[3]{{{{x}^{5}}{{y}^{{12}}}}}={{x}^{{\frac{5}{3}}}}{{y}^{{\frac{{12}}{3}}}}={{x}^{{\frac{3}{3}}}}{{x}^{{\frac{2}{3}}}}{{y}^{4}}=x\cdot {{x}^{{\frac{2}{3}}}}{{y}^{4}}=x{{y}^{4}}\sqrt[3]{{{{x}^{2}}}}$. Pretty cool!**Combine any like terms**. If you’re adding or subtracting terms with the same roots and/or variables, you can put these together. Almost think of a**radical expression**(like $ \sqrt{2}$)**like another variable**. Example: $ 4{{x}^{2}}\sqrt{2}-2{{x}^{2}}\sqrt{2}=2{{x}^{2}}\sqrt{2}$.**Rationalize radical fractions**, which means getting rid of radicals in the denominator.

Now let’s put it altogether. Here are some (difficult) examples; just remember that you have to be really, really careful doing these! Make sure the answers have no negative exponents.

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

**Note**: We’ll see more of these types of problems here in the** Solving Radical Equations and Inequalities **section. Some of the more complicated problems involve using **Quadratics**.

Now we’ll try solving equations involving exponents and roots. **Generally, to get rid of the exponents, take radicals of both sides, and to get rid of radicals, 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**. **Thus, 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**.

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. When we have **fractional** **exponents**, the radical is **even** when the **numerator **is even.

Also, when the original problem contains an **even root sign**, we need to check answers to make sure there are** no negative numbers under the even root sign** (no negative radicands). We also must check answers when we **raise both sides to an even exponent** (for example, **square both sides**).

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 here in the** Solving Radical Equations and Inequalities **section.

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

You can also check your answers using a **graphing calculator** by putting in what’s on the left of the = sign in “$ {{Y}_{1}}=$” and what’s to the right of the equal sign in “$ {{Y}_{2}}=$”. 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, using an example from above:

## Solving 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 must make sure our answer takes into account what we call the **domain restriction**: we must make sure what’s **under an even radical** is **0 or** **positive**, so we may have to create another inequality. To do this, we’ll set what’s under the even radical to greater than or equal to **0**, solve for $ x$, Then we 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; these are pretty straightforward, **since we know the sign of the values on both sides, so we can square both sides safely**. It gets trickier when we don’t know the sign of one of the sides.

**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.

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On to **Introduction to Multiplying Polynomials** – you are ready!