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

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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 again and again by the number in the exponent. So if you have \({{2}^{3}}\), you just have \(2\times 2\times 2\). It’s that easy! It just means that you multiply the number “2” by itself 3 times. Another way to describe \({{2}^{3}}\) is “2 to the third power” or “2 cubed”.

You may have heard the expression “to square” a number, or “the square” of a number. This just means you raise it to the 2nd power (the exponent is 2) or multiply it by itself. So if we square 4, we have \({{4}^{2}}=16\).

One example of raising a number to the 3rd power (or multiplying it by itself 3 times) is a Rubik’s Cube. We can figure out how many little cubes (little boxes) 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. So we can multiply 3 by itself 3 times to get the total number of cubes: \(3\,\times 3\times 3={{3}^{3}}=27\). So if we took a Rubik’s Cube apart, we would have 27 little cubes – can you see that?

Since know you know what an exponent is, we can revisit finding prime factors of a number for a minute. When we found the prime factors of 12, we got \(\displaystyle 12=2\times 2\times 3\). Now we can rewrite it with exponents (which is how it’s usually done); this would be \(\displaystyle 12={{2}^{2}}\times 3\) or \(\displaystyle 12=\left( {{{2}^{2}}} \right)\left( 3 \right)\).

I’m not going to go into a lot of detail here, but I also want to mention that you can have exponents that aren’t positive — exponents of 0, and negative exponents. These concepts are a little weird, but sometimes math works in mysterious ways. Just accept them as the truth for now (as you accept that your shoe size is now a size 6, or whatever it is!) and we’ll talk more about the why’s later.

First of all, any number with an exponent of 0 is 1:

\({{\text{(any number)}}^{0}}=\,{{5}^{0}}={{10000000000000}^{0}}={{(-48384304)}^{0}}=1\)

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.

Now if we raise a number to a negative exponent, it’s 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 when we talk about the Algebra sections, but here are some examples:

 \(\displaystyle {{2}^{{-2}}}=\frac{1}{{{{2}^{2}}}}\,\,\left( {\text{or }\frac{{{{1}^{2}}}}{{{{2}^{2}}}}} \right)=\frac{1}{4}\)  \(\displaystyle {{\left( {\frac{1}{2}} \right)}^{{-2}}}={{\left( {\frac{2}{1}} \right)}^{2}}=\frac{{{{2}^{2}}}}{{{{1}^{2}}}}=\frac{4}{1}=4\)  \(\displaystyle {{\left( {\frac{3}{4}} \right)}^{{-3}}}={{\left( {\frac{4}{3}} \right)}^{3}}=\frac{{{{4}^{3}}}}{{{{3}^{3}}}}=\frac{{4\times 4\times 4}}{{3\times 3\times 3}}=\frac{{64}}{{27}}\)

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 numbers 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.) So 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 together 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  \(\sqrt[3]{{-27}}=-3\)

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 \(\sqrt{{16}}=4\) (thus 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. (We’ll talk more about these numbers in the Introduction to Algebra section.)
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), such as \(\sqrt{{-4}}\). This is because we can’t multiply two numbers together to get a negative number — try it yourself!  We’ll talk about these different kinds of numbers in the Introduction to Algebra 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 \(\displaystyle \frac{1}{3}\). So you can see the pattern here.  This is a little weird, but it’s just something you’ll want to remember.

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:

Exponent and Radical Rules Example Notes

\(\begin{align}{{\left( {\text{negative number}} \right)}^{{\text{odd}}}}&=\,\text{negative}\\{{\left( {\text{negative number}} \right)}^{{\text{even}}}}&=\,\text{positive}\\-{{\left( {\text{any number}} \right)}^{{\text{even}}}}&=\,\text{negative}\\\sqrt[{\text{odd}}]{{\text{negative number}}}&=\,\text{negative}\end{align}\)

\(\begin{array}{c}{{\left( {-2} \right)}^{3}}=-8;\,\,\,\,-{{2}^{3}}=-8\\{{\left( {-2} \right)}^{4}}=16;\,\,\,\,\,-{{2}^{4}}=-16\\\sqrt[3]{{-27}}=-3\end{array}\) A negative number raised to an even exponent is always positive, but be careful about including parentheses in that case. A negative sign before the number makes the whole expression negative.

 

For odd exponents and negative numbers, it doesn’t matter.

\(\sqrt[{}]{{\text{number}}}\) is the same as \(\sqrt[2]{{\text{number}}}\)

\(\sqrt{4}=2\text{ means }\sqrt[2]{4}=2\)

When we take a square root, it’s like there’s an invisible 2 in the root.
 \({{\left( {\text{number}} \right)}^{0}}=1\)  \({{\left( {\text{437434729}} \right)}^{0}}=1\) Keep calm and trust me on this one. Don’t try to figure it out.

\(\begin{array}{l}\sqrt[{}]{0}=0\\\sqrt[{}]{1}=1\end{array}\)

 \(\begin{array}{c}0\,\,\times \,\,0=0\\1\,\,\times \,\,1=1\end{array}\)

You can take the square root of zero and one, but not an even square root of a negative number.

\(\sqrt[{\text{even} }]{{\text{negative number}}}\) doesn’t exist for real numbers

\(\begin{array}{c}\sqrt[4]{{-16}}\\\,\,\,\,\,\,=\text{ no real solution}\end{array}\)

We can never multiply two (or four, and so on) numbers together to get a negative number (try it!). (Later we’ll see that this is an “imaginary number”.)

\(\begin{array}{c}\sqrt{{\text{positive number}\,\,\text{squared}}}=\,\,\text{that number}\\\\\sqrt[3]{{\text{number}\,\,\text{cubed}}}=\,\,\text{that number}\\\\\sqrt[{\text{even} }]{{{{{\left( {\text{positive number}} \right)}}^{{\text{even}}}}}}=\text{the number}\\\\\sqrt[{\text{even} }]{{{{{\left( {\text{negative number}} \right)}}^{{\text{even}}}}}}=\left| {\text{the number}} \right|\end{array}\)

 \(\begin{array}{c}\sqrt{{{{5}^{2}}}}=5\\\\\sqrt[3]{{{{5}^{3}}}}=5\\\\\sqrt{{{{{\left( {-5} \right)}}^{2}}}}=5\end{array}\)

If we square a negative number, it will turn into a positive number, and the square root of that will be positive. (We don’t have to worry about this with odd roots, since they can be negative).

 

For example, \(\sqrt[{}]{{25}}=5\), even though \(-5\,\times -5=25\). Think of taking the absolute value of the original number when you take the even root.

                  \(\begin{array}{l}\sqrt{{\text{positive number}}}\,\,\times \,\,\sqrt{{\text{positive number}}}\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=\text{ that number}\end{array}\)

 \(\sqrt{5}\times \sqrt{5}=5\)

Multiplying two square roots together sort of undoes the square root.
\(\begin{array}{l}\sqrt{{\text{positive number} }}\,\,\,\times \,\,\sqrt{{\text{positive number} }}\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=\,\,\sqrt{{\text{product of the numbers}}}\end{array}\)

 \(\sqrt{5}\times \sqrt{6}=\sqrt{{30}}\)

\(\displaystyle \frac{{\sqrt{{30}}}}{{\sqrt{6}}}=\sqrt{{\frac{{30}}{6}}}=\sqrt{5}\)

This works for division, too. Note that his does not work for adding and subtracting numbers under the root sign: \(\sqrt{5}+\sqrt{6}\ne \sqrt{{11}}\).

 \(\sqrt{{{{{\left( {\text{number}} \right)}}^{{\text{exponent}}}}}}={{\left( {\sqrt[{}]{{\text{number}}}} \right)}^{{\text{exponent}}}}\)

\(\displaystyle \sqrt{{{{3}^{2}}}}={{\left( {\sqrt{3}} \right)}^{2}}=\sqrt{9}=3\) When you raise a number inside a root to a power, it doesn’t matter if you raise it inside or outside the root.

\(\displaystyle \begin{array}{l}\sqrt[{\text{root}}]{{\text{numbe}{{\text{r}}^{{\text{exponent}}}}}}={{\left( {\sqrt[{\text{root}}]{{\text{number}}}} \right)}^{{\text{exponent}}}}\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,={{\left( {\text{number}} \right)}^{{\frac{{\text{exponent}}}{{\text{root}}}}}}\end{array}\)

\(\begin{array}{l}\sqrt[4]{{{{{\left( 9 \right)}}^{2}}}}={{\left( {\sqrt[4]{9}} \right)}^{2}}={{9}^{{\frac{2}{4}}}}\\\,\,\,\,\,\,\,={{9}^{{\frac{1}{2}}}}=\sqrt{9}=3\end{array}\)

You can take an expression in radical form and turn it into rational form by using a fractional exponent with the original exponent on top and root on bottom (remember: the root is in a “cave” so it needs to go on the bottom). Try it with your calculator!

\(\displaystyle \begin{array}{l}{{\left( {{{{\left( {\text{number}} \right)}}^{{\text{exponent}}}}} \right)}^{{\text{exponent}}}}\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,={{\left( {\text{number}} \right)}^{{\text{exponent}\,\,\,\times \,\,\,\text{exponent}\,}}}\end{array}\)

\(\displaystyle {{\left( {{{2}^{2}}} \right)}^{3}}={{2}^{{\left( {2\,\,x\,\,3} \right)}}}={{2}^{6}}=64\)

When we raise a number to an exponent, and then raise that to another exponent, it’s just like raising the original number to the product (multiply) of those two exponents.

\(\displaystyle \text{numbe}{{\text{r}}^{{\text{negative }}}}=\frac{1}{{\text{numbe}{{\text{r}}^{{\text{make positive}}}}}}\)

 

\(\displaystyle \text{fractio}{{\text{n}}^{{\text{negative}}}}={{\left( {\text{reciprocal}} \right)}^{{\text{make positive}}}}\)

\(\displaystyle {{4}^{{-3}}}=\frac{1}{{{{4}^{3}}}}=\frac{1}{{64}}\)

\(\displaystyle {{\left( {\frac{3}{4}} \right)}^{{-2}}}={{\left( {\frac{4}{3}} \right)}^{2}}=\frac{{16}}{9}\)

When we raise a number or fraction to a negative exponent, it has nothing to do with something being negative. Instead put a 1 over the number, or flip the fraction (reciprocal) and then make the negative exponent positive.

 

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

Sorry – I can’t leave a radicals discussion without a brief introduction to more advanced topics that you’ll see again in the Algebra section.

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.)

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Now that we have two different numbers with \(\sqrt{2}\) in them, we can actually combine them. 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:

\(\displaystyle \color{#800000}{{\frac{4}{{3\sqrt{2}}}}}=\frac{4}{{3\sqrt{2}}}\cdot 1=\,\frac{4}{{3\sqrt{2}}}\cdot \frac{{\sqrt{2}}}{{\sqrt{2}}}=\frac{{4\sqrt{2}}}{{3\sqrt{2}\cdot \sqrt{2}}}=\frac{{4\sqrt{2}}}{{3\cdot 2}}=\frac{{4\sqrt{2}}}{6}=\frac{{2\sqrt{2}}}{3}\)

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

Putting numbers in Scientific Notation is something you’ll see in both your math and science classes. “Regular numbers” are called Standard Notation.

Sometimes numbers are too big or too small to put in calculators or to use in studies or research. Scientific notation is the answer to this problem and it’s easy to go back and forth between “regular” numbers and scientific notation (which are just numbers multiplied by powers of 10) if we just remember the rules.

Think of scientific notation as “abbreviating” numbers by putting numbers between 1 and 10 together and 10 raised to a number together.

Here’s an example.  Let’s say you’ve read that your favorite singer is filthy rich and has 9.8 million dollars, or $9,800,000 (I know; it’s ridiculous they make so much!).  Which way is easier to write what they are worth: $9.8 million, or $9,800,000?  See how the first way is much easier?

So we use a number between 1 and 10 (not including 10; for example 9.9999999… would work) and multiply it by 10 raised to a number. 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 we had to use a positive power of 10 to make up for that.

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 had to use a negative power of 10 to make up for that.

HINT:  When you are 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:

Math Notes
 \(3500=3.5\times {{10}^{3}}\) 3.5 is between 1 and 10; we went from a larger number to smaller number, so we used a positive exponent.
 \(.0000001=1\times {{10}^{{-7}}}\) 1 is between 1 and 10; we went from a smaller number to larger number, so we used a negative exponent.
 \(.123456=1.23456\times 1{{0}^{{-\,1}}}\) 1.23456 is between 1 and 10; we went from a smaller number to larger number, so we used a negative exponent.
 \(1000000=1\times {{10}^{6}}\) 1 is between 1 and 10; we went from a larger number to smaller number, so we used a positive exponent.

 

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 you are 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:

Math Notes
 \(3.5436\times {{10}^{{10}}}=35436000000\) Since we have a positive exponent, we need to end up with a larger number, so we move the decimal 10 places to the right.
 \(2.34\times {{10}^{{-4}}}=.000234\) Since we have a negative exponent, we need to end up with a smaller number, so we move the decimal 4 places 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!

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