# Powers, Exponents, Radicals (Roots), and Scientific Notation

## 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 3nd 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 3rd 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 (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 sectionbut 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.

Some examples:

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!

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