If you have worked through the scientific notation tutorial then you have already studied a very specific example of using exponents. In this tutorial you will be exploring the use of exponents in a more general way.
Astronomy uses a great many equations that involve exponents, as do other branches of science and engineering. Although the number of equations that are introduced in this course are restricted to a handful of the most fundamental ones, many of the equations, most in fact, will contain exponents in them.
Your instructor may or may not require that you understand all of the details in this tutorial. It largely depends on whether or not your instructor wants you to calculate the actual values of quantities in scientific notation. You may want to get clarification regarding your instructor’s expectations for the course. However, regardless of the depth to which your instructor wants to discuss the equations, you should feel comfortable with the first two parts of this tutorial.
A caveat: In this tutorial we won’t worry about the situation where negative numbers are raised to exponents, such as \((-3)^{1/2}.\) Problems like these are not necessary for our purposes and can lead to extra complications beyond the scope of the text for this course.
Sections
- The Basics
- Determining Roots
- Multiplying and Dividing Numbers With the Same Base
- Adding Numbers Together in Scientific Notation
- A Final Example
- Summary of Operations Involving Exponents
- Quiz Yourself
- Answers
The Basics
The most important fact that you need to know about exponents is that integer exponents are nothing more than a shorthand for describing how many times a number or variable is multiplied by itself.
In the scientific notation tutorial you learned how to interpret numbers like \( 3.14159 \times 10^9 \). The \(10^9 \) simply means that you multiply \(10\) by itself \(9\) times, or
$$10^9 = 10 \times 10 \times 10 \times 10 \times 10 \times 10 \times 10 \times 10 \times 10 = \text{1,000,000,000} $$
which is a one with nine zeros after it, or one billion. Overall, \(3.14159 \times 10^9 = \text{3,141,590,000}\).
For a number in scientific notation with a negative exponent such as \(3.14159 \times 10^{-9}\), the \(10^{-9}\) means that you divide one by \(10^9\), or
$$10^{-9} = \frac{1}{10^9} = \frac{1}{10 \times 10 \times 10 \times 10 \times 10 \times 10 \times 10 \times 10 \times 10} = \frac{1}{\text{1,000,000,000}} = 0.000\,000\,001$$
which is eight zeros followed by \(1\) (nine digits in all) to the right of the decimal point. Overall, \(3.14159 \times 10^{-9} = 0.000\,000\,003\,141\,59\).
Any number raised to an integer power works just like \(10\) raised to an integer power. For example
$$3^4 = 3 \times 3\times 3\times 3 = 81$$
and
$$3^{-4} = \frac{1}{3^4} = \frac{1}{81}$$
Since the fraction
$$\frac{1}{81}$$
is exactly the same as saying \(1 \div 81\), we find that \(3^{-4} = 0.012\,34\). (Note that if you don’t feel comfortable in working with fractions you may want to review the fractions tutorial.)
In equations, rather than explicitly writing numbers raised to some power, variables are used instead as placeholders for any number that you might need to put in the equation. (Be sure to review the variables tutorial if you are uncomfortable with the step of abstraction in working with variables rather than numbers.)
Suppose that you run into an equation that has \(P^2\) in it (and you will by the way!), this statement is exactly the same as saying \(P \times P\). If \(P\) for a particular case happens to be \(3\), then \(P \times P = 3 \times 3 = 9\). Similarly, if \(P = 12.2\), then \(P^2 = 12.2 \times 12.2 = 148.84\).
The situation is similar for any variable raised to a positive integer. For example, if \(a = 6.3\) then \(a^3 = 6.3 \times 6.3 \times 6.3 = 250.047\).
Variables with negative exponents behave the same way, except that the operation includes dividing \(1\) by the variable raised to the positive power (this is exactly what happens in scientific notation when \(10\) is raised to a negative integer). For example,
$$r^{-2} = \frac{1}{r^2} = \frac{1}{r \times r}.$$
If, in a specific case \(r = 8.3,\) then
$$r^{-2} = 8.3^{-2} = \frac{1}{8.3 \times 8.3} = 0.01452.$$
Of course calculators and computers are just as capable in computing numbers raised to negative powers as they are with computing numbers raised to positive powers. Because of this, the intermediate steps you have seen above are rarely necessary, but they do help you to understand the meaning of what you are asking your calculator to do. You should always understand what is happening when you ask your calculator to carry out an operation. You should also have an approximate idea of what the answer should be!
- What is the value of \(3.2^6\) ?
- What is the value of \(3.2^{-6}\) ?
- If \(a = 5\) and \(b = 3\), determine the value of \(a^b\) ?
- If \(a = 5\) and \(b = -3\), determine the value of \(a^b\) ?
Determining Roots
There are times when it becomes necessary to perform a mathematical operation that is essentially the reverse of raising a number or a variable to a power; that is, to determine the root of a number.
You are likely familiar with the idea of taking the square root of a number. Determining the square root is the process that answers the question: What number times itself will give me the number I already have? For the number \(4\), the answer to the question is obviously \(2\). In other words \(2 \times 2 = 2^2 = 4\).
Symbolically, the square root operation is indicated by \(\sqrt{\ \ \ }\). However, in parallel with the notation of an exponent, a square root is also sometimes expressed as \((\ )^{1/2}\). (Why this exponential notation is equivalent mathematically to the square root symbol will be explained in the section Multiplying and Dividing Numbers With the Same Base.) Using the more common notation, taking the square root of \(4\) and getting the result, \(2\), is written as
$$\sqrt{4} = 2.$$
Similarly, the square root of \(16\) is \(4\), since \(4 \times 4 = 4^2 = 16\).
$$\sqrt{16} = 4$$
It is also possible to determine results for the square roots of numbers that don’t have simple integer answers (e.g., \(1, 2, 3, \ldots)\). This is easily done today with calculators since virtually all calculators have a square root operation built into them. For example,
$$\sqrt{3} = 1.732\ldots\qquad\text{and}\qquad\sqrt{937} = 30.610\ldots.$$
The trailing dots (\(\ldots\)) indicate that there are more (in fact infinitely more) digits in the answer, but three digits after the decimal point is certainly plenty for our purposes in this course. Using the exponential notation for the square root,
$$3^{1/2} = 1.732\ldots\qquad\text{and}\qquad937^{1/2} = 30.610\ldots.$$
Of course there are other roots possible other than the square root. In fact, it is possible to take cubed roots, fourth roots, fifth roots, or even \(3.14159\)th roots; whatever your heart desires. Again modern scientific calculators come to the rescue, but different calculators use varying button notations. It is common for calculators to have buttons such as
$$\sqrt[y]{x}\qquad\text{or}\qquad x^y\qquad\text{or}\qquad{\Large\hat{}}.$$
You should make sure that you know how your particular calculator is used to calculate roots of numbers. If you aren’t sure, you should ask your instructor or someone who is familiar with your particular brand of calculator.
By now you have probably realized that if you raise a number to a particular power, and then take the root of the result using the value of the exponent, that you get the number back that you started with. In other words,
$$\sqrt[5]{3^5} = \sqrt[5]{243\vphantom{3^5}} = 3\qquad\text{or}\qquad\left(3^5\right)^{1/5} = (243)^{1/5} = 3.$$
The two operations are inverses of each other. It is also the case that if you determine the root first and then raise that result to the equivalent exponent, you again get your starting value back. For example,
$$\left(\sqrt[5]{3}\right)^5 = 3\qquad\text{or}\qquad(3^{1/5})^5 = 3.$$
In general, using variables as placeholders for any number,
$$\sqrt[y]{x^y} = x,\qquad\left(x^y\right)^{1/y} = x,\qquad\left(\sqrt[y]{x}\right)^y = x,\qquad\text{and}\qquad(x^{1/y})^y = x.$$
How is all of this useful in this course? Suppose you know that \(P^2 = 8\), but you are asked to find \(P\) itself? In this case it isn’t obvious what the correct answer is. You know that \(2 \times 2 = 4\) and \(3 \times 3 = 9\), so the answer must be somewhere between \(2\) and \(3\), but how do you find it? Since \(\sqrt{P^2} = P\), you need only take the square root of \(\sqrt{P^2}\), or \(\sqrt{8} = 8^{(1/2)} = 2.828\).
If you know that \(a^3 = 12\), then to determine \(a\) you must take the cubed root of \(12\), or \(\sqrt[3]{12} = 12^{1/3} = 2.289\).
Multiplying and Dividing Numbers With the Same Base
You are often asked to multiply two numbers together that have the same base (the same number that an exponent is applied to). This is particularly common when you are using scientific notation and the base is \(10\).
What is \(10^3 \times 10^5\)? Expanding the exponents you see that
$$10^3 \times 10^5 = (10 \times 10 \times 10) \times (10 \times 10 \times 10 \times 10 \times 10) = 10^8.$$
As you can see, the end result of \(10^3 \times 10^5\) was just \(10^{3 + 5} = 10^8\). All you need to do is add the exponents.
The same is true for any two numbers multiplied together that have the same base. For example,
$$3^4 \times 3^2 = (3 \times 3 \times 3 \times 3) \times (3 \times 3) = 3^6.$$
In general this means that
$$x^a \times x^b = x^a\,x^b = x^{a + b}.$$
[Recall that the \(\times \) symbol is implied when you have variables, or a number in front of a variable, up against each other without an explicit symbol. The same is true if you have quantities in parentheses up against one another. For example, in the previous numerical case,
$$3^4 \times 3^2 = 3^4\,3^2 = (3^4)\,(3^2) = (3 \times 3 \times 3 \times 3)\times (3 \times 3) = 3^6 = 729.$$
What happens when one or both of the exponents is negative? The same thing. As an example
$$10^4 \times 10^{-2} = (10 \times 10 \times 10 \times 10) \times \frac{1}{10 \times 10} = 10 \times 10 = 10^2. $$
In this example \(10^4 \times 10^{-2} = 10^{(4 – 2)} = 10^2 = 100.\) The general statement that \(x^a \times x^b = x^a\,x^b = x^{a + b}\) applies whether \(a\) and/or \(b\) are positive or negative.
For the special case of writing numbers in scientific notation remember that the notation really is representing the mathematical process of using exponents. Also remember that multiplication is commutative, or \(x \times y = y \times x\).
As an example of multiplying two numbers in scientific notation:
$$(3.14159 \times 10^9) \times (4.235 \times 10^2) = (3.14159 \times 4.235) \times \left(10^9 \times 10^2\right)\\ = 13.304 \times 10^{11} = \left(1.3304 \times 10^1\right) \times 10^{11} = 1.3304 \times 10^{12}.$$
You should make sure that you totally understand each of the steps just shown.
Another important idea involving exponents is that if you multiply two numbers together and then raise the product to a power, it is exactly the same as first raising each number to that power and then multiplying the two results together. To illustrate this, consider \((3 \times 2)^4\):
$$(3 \times 2)^4 = 6^4 = 1296.$$
But this is also the same as
$$(3 \times 2)^4 = 3^4 \times 2^4 = 81 \times 16 = 1296.$$
Symbolically (i.e., using variables),
$$(xy)^a = x^a\,y^b.$$
Finally, consider what happens when you raise a number to an exponent when the number itself is already raised to another exponent. As an example, how do you treat \( (3^2)^3\)? First, remember that some number raised to a power is just that number times itself the same number of times as the exponent indicates, or \(\small a^3 = a \times a \times a\). Applying that same rule to our example,
$$(3^2)^3 = 3^2 \times 3^2 \times 3^2 = 3^{(2 + 2 + 2)} = 3^6 = 729.$$
More concisely, \((3^2)^3 = 3^{2 \times 3} = 3^6 = 729\).
Writing the behavior symbolically,
$$(x^a)^b = x^{(a\times b)} = x^{ab}.$$
Mentioned in the last section is the idea of finding roots of numbers. In that section it was pointed out that there are two ways to represent the operation of calculating roots: (\sqrt[y]{x}) and (\small x^{1/y}). It was also pointed out that if you take the root of some number and then raise the result to that associated exponent, the initial number results, or (\left[(x)^{1/a}\right]^a = x.) Using what you have just learned, this is now simple to understand since:
$$\left(x^{1/a}\right)^a = x^{\left(\frac{1}{a}\right)\times a} = x^{\left(\frac{a}{a}\right)} = x^1 = x.$$
To see how this rule plays out when using scientific notation, you need to consider the ideas
that \((xy)^a = x^a\,y^a\) and \((x^a)^b = x^{ab}\) together. For example, what is \((3.14159 \times 10^9)^3\)?
$$(3.14159 \times 10^9)^3 = (3.14159)^3 \times (10^9)^3\\ = 3.14159^3 \times 10^{(9 \times 3)} = 31.0062 \times 10^{27} = 3.10062 \times 10^{28}$$
Don’t forget that the proper form of scientific notation requires that the decimal point always be to the right of the first non-zero number, hence the last step above.
Adding Numbers Together in Scientific Notation
Consider what happens when you need to add (or subtract) two numbers in scientific notation. The critical rule to remember is: In order to add the two numbers together in scientific notation, we must first write them with the same power of 10.
As an example, what happens when you add \(3.14159 \times 10^9\) with \(8.245 \times 10^9\)? In this case the powers
of \(10\) are already the same, so applying the distributive property of algebra, \((a + b)c = ac + bc\), we find
$$3.14159 \times 10^9 + 8.245 \times 10^9 = (3.14159 + 8.245) \times 10^9\\ = 11.38659 \times 10^9 = 1.138659 \times 10^{10}.$$
But what happens when the two numbers don’t have the same power of ten? You must write one of the numbers so that they do have the same powers of \(10\) before you can add them together, but note that it doesn’t matter which one you change. (This requires temporarily violating the standard for scientific notation that the first non-zero number must be to the left of the decimal point.) For example,
$$3.14159 \times 10^9 + 8.245 \times 10^7 = 3.14159 \times 10^9 + 8.245 \times \left(10^{-2} \times {10}^9\right)\\ = 3.14159 \times 10^9 + \left(8.245 \times 10^{-2}\right) \times 10^9\\ = 3.14159 \times 10^9 + 0.08245 \times 10^9 = 3.22404 \times 10^9,$$
or, changing the other number,
$$3.14159 \times 10^9 + 8.245 \times 10^7 = 3.14159 \times \left(10^2 \times 10^7\right) + 8.245 \times 10^7\\ = \left(3.14159 \times 10^2\right) \times 10^7 + 8.245 \times 10^7\\ = 314.159 \times 10^7 + 8.245 \times 10^7 = \left(314.159+8.245\right) \times 10^7\\ = 322.404 \times 10^7 = 3.22404 \times 10^9.$$
A Final Example
Note that this problem has numerous steps, but none of them involve calculations that you haven’t seen before. The “trick” is to take it one step at a time, much like figuring out how you are going to get from one place to another in a city; which streets you need to be on and traveling in which direction.
As one final example, consider the expression \((m_1 + m_2)P^2\), where \(m_1 = 5.9736 \times 10^{24}\), \(m_2 = 1.9891 \times 10^{30}\), (the subscripts \(1\) and \(2\) are used to distinguish between two different objects), and \(P = 3.1558 \times 10^7\). [In case you’re curious, the expression is actually a portion of an equation known as Kepler’s Third Law, where the values are \(m_1\), the mass of Earth (in kg), \(m_2\), the mass of the Sun (in kg), and \(P\), the orbital period of Earth around the Sun (in s).]
Substituting the numbers into the expression,
$$(m_1 + m_2)P^2 = (5.9736 \times 10^{24} + 1.9891 \times 10^{30}) \times (3.1558 \times 10^7)^2.]
Before fully evaluating the expression, look carefully at the numbers being added together in the first set of parentheses; they don’t have the same exponent for \(10\). In fact, the mass of Earth \(m_1\) in the expression) is much smaller than \(m_2\), the mass of the Sun. Recalling that in order to add two numbers together in scientific notation we must first write them both with \(10\) raised to the same power, we find
$$5.9736 \times 10^{24} + 1.9891 \times 10^{30} = 5.9736 \times (10^{-6} \times 10^{30}) + 1.9891 \times 10^{30}\\ = 0.000\,005\,9736 \times 10^{30} + 1.9891 \times 10^{30}\
Again, recalling the distributive property of algebra, \((a + b)c = ac + bc\), we find
$$5.9736 \times 10^{24} + 1.9891 \times 10^{30} = (0.0000059736 + 1.9891)\times 10^{30}\\ = 1.9891059736 \times 10^{30} \approx 1.9891 \times 10^{30}.$$
which is just the value of \(\small m_2\), the mass of the Sun. (The symbol \(\approx\) means approximately equal to.) Since the mass of the Sun is so much larger than the mass of Earth, adding the mass of Earth to the mass of the Sun makes very little difference.
Finally, going back to the original expression, since multiplication takes precedence over addition (see the order of operations tutorial), and since we must evaluate the quantities in parentheses before multiplying the results together, we arrive at
| \((m_1 + m_2)P^2\) | = | \(\left(5.9736 \times 10^{24} + 1.9891 \times 10^{30}\right) \times \left(3.1558 \times 10^7\right)^2\) |
| = | \(\left(1.9891 \times 10^{30}\right)\times \left(3.1558 \times 10^7\right)^2\) | |
| = | \(\left(1.9891 \times 10^{30}\right) \times (3.1558)^2 \times \left(10^7\right)^2\) | |
| = | \(\left(1.9891 \times 10^{30}\right)\times \left(9.959 \times 10^{14}\right)\) | |
| = | \(\left(1.9891 \times 9.959\right) \times \left(10^{30} \times 10^{14}\right)\) | |
| = | \(19.8096 \times 10^{44}\) | |
| = | \(1.98096 \times 10^{45}\). |
As mentioned previously, all of this arithmetic can be done quickly using calculators without ever needing to carry out the intermediate steps, but it is very important to understand what is going on under the hood of your calculator if you are to understand the answer your calculator ultimately gives you.
A Summary of Operations Involving Exponents
A great deal of information and numerous examples were presented in this tutorial. As a result, it is worthwhile putting all of the information in one place for reference.
[By the way, it is important to read and interpret equations, just like you would with any language. For example, why does \((xy)^a = x^ay^a\)? It is because \((xy)^a = (xy) \times (xy) \times (xy)\times \ldots \times (xy)\), where \(xy\) is multiplied by itself \(a\) times. This means that \(x\) and \(y\) are both multiplied \(a\) times.]
| Expression | Equivalent | |
|---|---|---|
| \(x^a\) | \(=\) | \(x\,x\,x\,x\ldots x\,x\ \ (a\text{ times})\) |
| \(x^{-a}\) | \(=\) | \(\frac{1}{x^a}\) |
| \(\sqrt[y]{x}\) | \(=\) | \(x^{1/y}\) |
| \((xy)^a\) | \(=\) | \(x^a\,y^a\) |
| \((x^a)^b\) | \(=\) | \(x^{ab}\) |
| \(\color{red}{(x + y)^a}\) | \(\color{red}\ne\) | \(\color{red}{x^a + y^a}\) |
Note that \(\ne\) means ”not equal to.”
The last expression is only included to remind you that the expression on the left DOES NOT equal the expression on the right!
Quiz Yourself
Try the following exercises to make sure that you really do understand the material just presented. By the way, computers and many calculators write scientific notation in “computer-speak,” where \(10\) raised to an integer (either positive or negative) replaces \(\times 10^\text{exponent}\) with \({\rm E(exponent)}\) or \({\rm e(exponent)}\); in other words, \(3.14159 \times 10^9\) is represented by \(3.14159\text{E}9\). If your calculator has the \(\Large{\hat{ }}\) symbol, then you would enter the number as \(3.14159 \times 10{\Large{\hat{ }}}9\). If your calculator has a \(10^x\) button then you would enter the number as \(3.14159\times9\,\,10^x\). If you have having problems with entering numbers in scientific notation, read the manual or help screens, or ask your instructor.
- What is the value of \(3.2^6\) ?
- What is the value of \(3.2^{-6}\) ?
- If \(a = 5\) and \(b = 3\), determine the value of \(a^b\) ?
- If \(a = 5\) and \(b = -3\), determine the value of \(a^b\) ?
- What is the value of \(\sqrt{13}\) ?
- Evaluate \(\sqrt[5]{13}\).
- What is the cubed root of \(925\), or \(\sqrt[3]{925}\) ?
- \(a = 2317\) and \(b = 5\). What is the value of \(\sqrt[b]{a}\)?
- Evaluate \(3^{1/8}\).
- \(a = \text{243,424}\) and \(b = 1/2342\). What does \(a^b\) equal?
- \(a = 2345\) and \(b = -4\). What is \(a^b\) equal to?
- What does \(\frac{1}{\sqrt[3]{8}}\) equal to?
- What is the value of \(7^4 \times 7^9\)?
- Calculate the product of two numbers written in scientific notation: \((3.938 \times 10^6) \times ({4.9325 \times 10}^{-6})\).
- What does \({({9.8765 \times 10}^{18})}^3\) equal?
- Evaluate \(\left(9.4^4\right)^{-1/8}\).
- Determine the value of \(3.9375 \times 10^4 + 9.345 \times 10^2\).
- Evaluate \(3.2459 \times 10^{-7} + 5.493 \times 10^{-9}\).
- What is the value of \(312.5 + 9.456 \times 10^8\)?
Answers
- \(\text{1073.74}\)
- \(0.000\,931\,323\)
- \(\text{125}\)
- \(0.008\)
- \(3.605\,55\)
- \(1.670\,28\)
- \(9.743\,48\)
- \(4.709\,61\)
- \(1.1472\)
- \(1.002\,35\)
- \(3.306\,96\times 10^{-14}\)
- \(0.5\)
- \(9.6889 \times 10^{10}\)
- \(1.942\,42\times 10^1\)
- \(9.634\,06\times 10^{56}\)
- \(0.326\,164\)
- \(4.030\,95\times 10^4\)
- \(3.300\,83\times 10^{-7}\)
- \(9.456 \times 10^8\)