Exponents and Powers | Scientific Notation |

Radical (Roots) | More Practice |

Simplifying and Rationalizing Radicals |

Note that we’ll talk about **Exponents and Radicals in Algebra** here.

## Exponents and Powers

Actually, I think students have difficulty with** powers**, or** exponents**, since they are so small; they really aren’t difficult. An **exponent** just means that you multiply that number (the **base**) again and again by the number in the exponent. For example, $ {{4}^{2}}=4\times 4=16$. It’s that easy! Another way to describe $ {{4}^{2}}$ is “**4** to the **second** **power**” or “**4 squared**”.

You may have also heard the expression “to **cube**” a number, or “the cube” of a number. This just means you raise it to the **3**nd power (the exponent is **3**) or multiply it by itself **3** times. If we cube **2**, we have $ {{2}^{3}}=2\times 2\times 2=8$.

A visual example of raising a base to the **3**rd power (or multiplying it by itself **3** times) is a Rubik’s Cube. We can figure out how many little square cubes are in the whole Rubik’s Cube by knowing that we have **3** cubes going across, **3** cubes going back, and **3** cubes going down. We can multiply **3** by itself **3** times to get the total number of cubes: $ 3\,\times 3\times 3={{3}^{3}}=27$. If we took a Rubik’s Cube apart, we would have **27** little cubes – can you see that?

Since now you know what an exponent is, we can revisit finding **Prime Factors **from the** Multiplying and Dividing **section. When we found the prime factors of **12**, for example, we got $ \displaystyle 12=2\times 2\times 3$. Now we can rewrite it with exponents (which is how it’s usually done) as $ \displaystyle 12={{2}^{2}}\times 3$ or $ \displaystyle 12=\left( {{{2}^{2}}} \right)\left( 3 \right)$.

Note that we can have exponents that aren’t positive: **exponents of 0**, and** negative exponents:**

Any base with an exponent of **0** is **1**:

$ {{\text{(any number)}}^{0}}=\,{{5}^{0}}={{1000}^{0}}={{(-4838)}^{0}}=1$

Also remember that **0** raised to any number except for **0** is just **0** (for example, $ {{0}^{{354}}}=0$). But **0** raised to **0** ($ {{0}^{0}}$) is undefined at this point. In Calculus, we see this sometimes, but for now, let’s say it’s undefined, and you won’t have any problems like this.

Raising a base to **a negative exponent** is the same as taking the reciprocal of that number (putting **1** over it if it’s not a fraction) and making the same exponent positive. We’ll get into this more in the **Exponents and Radicals in Algebra **section, but here are some examples:

Notice that when we remove the parentheses of a fraction raised to an exponent, the exponent goes to both the top (numerator) and bottom (denominator) – I like to call it “pushing it through” the fraction.

One other thing — be careful when raising **negative bases to powers**. You have to think about when the negative number is **inside **the exponent and when it’s not. So $ -{{2}^{2}}=-(2\times 2)=-4$, but $ {{\left( {-2} \right)}^{2}}=-2\times -2=4$. (Remember that a negative number times a negative number is a positive number.) Thus, when a negative number is raised to an **even power**, it always turns positive. When a negative number is raised to an **odd power**, it stays negative. We’ll talk about this more when we talk about **Order of Operations** in the next section.

Note that you can also raise decimals to exponential powers; for example, $ {{2.1}^{2}}=4.41$.

## Radicals (Roots)

**Radicals **(also called **roots***)* are what we get when we work backwards from raising a number to an exponent; they are how many times a number is multiplied by itself to get a number. For example, the **square** root of **16** is 4, since $ 4\times 4=16$ (we multiplied **4 **by itself **two** times). Again, think of radicals as the “undoing” of raising numbers to powers.

You write a radical with a funny sign that almost looks like a division: $ \sqrt{{16}}=4$. We’ll see later that there is an invisible “**2**” inside the square root sign ($ \sqrt[2]{{16}}=4$), since we are finding **two** numbers multiplied together that equal **4**. If we are finding **3** numbers multiplied together, we are taking what we call the **cube root** of a number and we put a little **3** in the root sign like this: $ \sqrt[3]{{27}}=3$ or $ \sqrt[3]{{-27}}=-3$. Remember that the root is on the outside (**3**, in this case), and what’s under the radical sign is the **radicand** (**27**, in this case).

Note that when we take **even** roots (like square roots), our answer is only the **positive** root, even though the negative root also works. When we take **odd** roots (like the cube root), the answer has **whatever sign is underneath the root sign**. Try multiplying back some numbers yourself to see why this is true. We’ll talk about this later in the **Exponents and Radicals in Algebra** section.

Some roots are **rational** and can be reduced to a real number, such as $ \sqrt{{16}}=4$ (**16** is called a **perfect square**), but most roots just won’t end up as a “good” number, or a number that has an exact answer. For example, if you put $ \sqrt{2}$ in a calculator, you get something like **1.4141213562**, but this is only an approximation, and it never really “resolves” itself. That’s why, for numbers like these where there is no exact root, your teacher will have you keep the radical in them. These numbers are called **irrational** since we can’t really get an exact answer with decimals or fractions. Also, some roots are actually not **real numbers** (numbers that are on the number line), but **imaginary**, meaning they don’t really exist, but you can do math with them. An example is $ \sqrt{{-4}}$, since we can’t multiply two numbers together to get a negative number — try it yourself! We’ll talk more about these different types of numbers in the **Types of Numbers and Algebraic Properties** section.

When we take the square root of a number, it’s the same thing as raising it to the $ \displaystyle \frac{1}{2}$. When we take the cube root of a number, it’s the same thing as raising it to the $ \displaystyle \frac{1}{3}$. Also, when we take the square root of a number raised to the third power, for example, this is the same as raising the number to the $ \displaystyle \frac{3}{2}$ power. These types of roots are in **rational form**, as opposed to **radical form**, such as $ \sqrt[2]{{{{4}^{3}}}}=\sqrt[{}]{{{{4}^{3}}}}={{\left( {\sqrt[{}]{4}} \right)}^{3}}$, since we’re displaying the root/exponent as a fraction.

More observations and a sum-up are below. Some of these concepts may be a little advanced and we will cover them again in the **Exponents and Radicals in Algebra** section, but I wanted to introduce them here:

Play around with these examples yourself and use other numbers. Again, we’ll talk more about exponents and radicals and how they work in the **Exponents and Radicals in Algebra **section**, **but I just wanted to give you an introduction.

In **Geometry**, you may have also use squares and cubes (raising a number to **3**) since we can use the concept to figure out areas and volumes of things (how big they are) — sort of like we did with the Rubik’s Cube.

## Simplifying and Rationalizing Radicals – an Introduction

Sometimes we have to **simplify radicals** and combine them in certain ways to make the math more “grammatically correct”. For example, suppose we are asked to simplify the following expression: $ \sqrt{2}+\sqrt{8}$

We can take perfect squares out from underneath the root sign with the **8** by factoring: $ \sqrt{8}=\sqrt{{4\times 2}}=\sqrt{4}\times \sqrt{2}=2\times \sqrt{2}=2\sqrt{2}$. See how we could “break up” the **8** and bring a **2** to the outside? (There are many more rules like this that we’ll see later.)

Now that we have two different numbers with $ \sqrt{2}$ in them, we can actually combine them; think of the $ \sqrt{2}$ almost like a variable. We have to put an invisible **1** in front of the first $ \sqrt{2}$ since we just have one of those. We have two of the other ’s so we have three total: $ \sqrt{2}+\sqrt{8}=\sqrt{2}+2\sqrt{2}=1\sqrt{2}+2\sqrt{2}=3\sqrt{2}$.

Another trick you’ll learn early on with roots is how to **rationalize** **denominators**. Again, it’s bad mathematical “grammar” to have a root in the denominator, so you need to multiply the top and bottom by the same root (which is **1**) to get it out of the denominator; for example:

$ \displaystyle \color{#800000}{{\frac{4}{{3\sqrt{2}}}}}\color{#5A5A5A}{=\,\frac{4}{{3\sqrt{2}}}\cdot \frac{{\sqrt{2}}}{{\sqrt{2}}}=\frac{{4\sqrt{2}}}{{3\sqrt{2}\cdot \sqrt{2}}}=\frac{{{{{\cancel{4}}}^{2}}\sqrt{2}}}{{3\cdot {{{\cancel{2}}}^{1}}}}=\frac{{2\sqrt{2}}}{3}}$

See how we ended up with no root in the denominator!

Again, if you don’t get all this at this point (before **Algebra**), don’t worry – you’ll get it later!

## Scientific Notation

So far, we’ve been using “regular numbers” or **Standard Notation**.** Scientific Notation**, on the other hand, is something you’ll see in both your math and science classes, and is a way to “abbreviate” very small and very large by numbers by multiplying numbers between **1** and **10** with (**10 raised to an exponent)**.

Here’s an example. Let’s say you’ve read that your favorite singer has **9.8** million followers on social media, or **9,800,000 **fans. Which way is easier to write: **9.8** million, or **9,800,000**? See how the first way is much easier?

With scientific notation, we use a number between **1** and **10** (not including **10**; but, for example, **9.999999** would work) and multiply it by **10** raised to a number (exponent). Then we have to count the number of decimal places that we moved the original decimal point to the new decimal point that is between **1** and **10**. If we moved the decimal point to the **right** (from a **smaller** number), we have a **negative** exponent. If we moved the decimal point to the **left** (from a **larger** number), we have a **positive** exponent.

For example, $ 9,800,000=9,800,000.0=9.8\times {{10}^{6}}$. This is because we moved the decimal place to the **left 6 places** (making the number smaller from **9,800,000** to **9.8**), and, to compensate, we need a **positive** power of **10. **Alternatively, $ .0056=5.6\times {{10}^{{-3}}}$. This is because we moved the decimal place to the **right 3 places** (making the number larger from **.0056** to **5.6**), and we need **a negative** power of **10** to compensate.

**Hint: **When converting a decimal to scientific notation, if you end up with a **larger** number (for example, **.004** to **4**), the power of **10** will be **negative**; if you end up with a **smaller** number (for example, **4000** to **4**), the power of **10 **will be **positive.**

More examples:

Sometimes we have to move back from scientific notation to a “regular number”, or standard notation. Notice that we will most likely need to add zeros, either at the end of the number, or after the decimal point, before the number starts, as shown below.

**Hint: **When converting from scientific notation back to a decimal, if you have a **positive** exponent, you need to make the first part of the number **larger**, so move the decimal to the** right. **If you have a** negative** exponent, you need to make the number **smaller**, so move the decimal to the **left**.

Make sure you understand how to go back and forth between scientific notation and the “regular” number! **Learn these rules and practice, practice, practice!**

**For Practice**: Use the **Mathway** widget below to try a **Power and Exponents** problem. Click on **Submit** (the blue arrow to the right of the problem) and click on **Describe the Transformation** 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 **Order of Operations PEMDAS** – you are ready!