Improve your math knowledge with free questions in 'Understanding exponents' and thousands of other math skills. The exponent of a number says how many times to use the number in a multiplication. In 8 2 the '2' says to use 8 twice in a multiplication, so 8 2 = 8 × 8 = 64. In words: 8 2 could be called '8 to the power 2' or '8 to the second power', or simply '8 squared'. 2020 GROMACS user survey is now live.The survey will help the GROMACS developers to prioritise future GROMACS developments. We would like input from researchers who perform any and all forms of molecular dynamics and whose experience using GROMACS ranges from zero experience to expert active users. Understanding exponents. As you saw in the video, exponents are written like this: 4 3 (you'd read it as 4 to the 3rd power). All exponents have two parts: the base, which is the number being multiplied; and the power, which is the number of times you multiply the base.
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Lesson 2: Exponents
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What are exponents?
Exponents are numbers that have been multiplied by themselves. For instance, 3 · 3 · 3 · 3 could be written as the exponent 34: the number 3 has been multiplied by itself 4 times.
Exponents are useful because they let us write long numbers in a shortened form. For instance, this number is very large:
1,000,000,000,000,000,000
But you could write it this way as an exponent:
1018
It also works for small numbers with many decimal places. For instance, this number is very small but has many digits:
.00000000000000001
It also could be written as an exponent:
10-17
Scientists often use exponents to convey very large numbers and very small ones. You'll see them often in algebra problems too.
Understanding exponents
As you saw in the video, exponents are written like this: 43 (you'd read it as 4 to the 3rd power). All exponents have two parts: the base, which is the number being multiplied; and the power, which is the number of times you multiply the base.
Because our base is 4 and our power is 3, we’ll need to multiply 4 by itself three times.
43 = 4 ⋅ 4 ⋅ 4 = 64
Because 4 · 4 · 4 is 64, 43 is equal to 64, too.
Occasionally, you might see the same exponent written like this: 5^3. Don’t worry, it’s exactly the same number—the base is the number to the left, and the power is the number to the right. Depending on the type of calculator you use—and especially if you’re using the calculator on your phone or computer—you may need to input the exponent this way to calculate it.
Exponents to the 1st and 0th power
How would you simplify these exponents?
71 70
Don’t feel bad if you’re confused. Even if you feel comfortable with other exponents, it’s not obvious how to calculate ones with powers of 1 and 0. Luckily, these exponents follow simple rules:
- Exponents with a power of 1
Any exponent with a power of 1 equals the base, so 51 is 5, 71 is 7, and x1 is x. - Exponents with a power of 0
Any exponent with a power of 0 equals 1, so 50 is 1, and so is 70, x0, and any other exponent with a power of 0 you can think of.
Operations with exponents
How would you solve this problem?
22 ⋅ 23
If you think you should solve the exponents first, then multiply the resulting numbers, you’re right. (If you weren’t sure, check out our lesson on the order of operations).
How about this one?
x3 / x2
Or this one?
2x2 + 2x2
While you can’t exactly solve these problems without more information, you can simplify them. In algebra, you will often be asked to perform calculations on exponents with variables as the base. Fortunately, it’s easy to add, subtract, multiply, and divide these exponents.
Adding exponents
When you’re adding two exponents, you don’t add the actual powers—you add the bases. For instance, to simplify this expression, you would just add the variables. You have two xs, which can be written as 2x. So, x2+x2 would be 2x2.
x2 + x2 = 2x2
How about this expression?
3y4 + 2y4
You're adding 3y to 2y. Since 3 + 2 is 5, that means that 3y4 + 2y4 = 5y4.
3y4 + 2y4 = 5y4
4r7 + 9r7
You can never add any of these as they’re written. This expression has variables with two different powers:
4r3 + 9r8
This one has the same powers but different variables, so you can't add it either:
4r2 + 9s2
Subtracting exponents
Subtracting exponents works the same as adding them. For example, can you figure out how to simplify this expression?
5x2 - 4x2
5-4 is 1, so if you said 1x2, or simply x2, you’re right. Remember, just like with adding exponents, you can only subtract exponents with the same power and base.
5x2 - 4x2 = x2
Multiplying exponents
Multiplying exponents is simple, but the way you do it might surprise you. To multiply exponents, add the powers. For instance, take this expression:
x3 ⋅ x4
The powers are 3 and 4. Because 3 + 4 is 7, we can simplify this expression to x7.
x3 ⋅ x4 = x7
What about this expression?
3x2 ⋅ 2x6
The powers are 2 and 6, so our simplified exponent will have a power of 8. In this case, we’ll also need to multiply the coefficients. The coefficients are 3 and 2. We need to multiply these like we would any other numbers. 3⋅2 is 6, so our simplified answer is 6x8.
3x2 ⋅ 2x6 = 6x8
You can only simplify multiplied exponents with the same variable. For example, the expression 3x2⋅2x3⋅4y2 would be simplified to 24x5⋅y2. For more information, go to our Simplifying Expressions lesson.
Dividing exponents
Dividing exponents is similar to multiplying them. Instead of adding the powers, you subtract them. Take this expression:
x8 / x2
Because 8 - 2 is 6, we know that x8/x2 is x6.
x8 / x2 = x6
What about this one?
10x4 / 2x2
If you think the answer is 5x2, you’re right! 10 / 2 gives us a coefficient of 5, and subtracting the powers (4 - 2) means the power is 2.
Raising a power to a power
Sometimes you might see an equation like this:
(x5)3
An exponent on another exponent might seem confusing at first, but you already have all the skills you need to simplify this expression. Remember, an exponent means that you're multiplying the base by itself that many times. For example, 23 is 2⋅2⋅2. That means, we can rewrite (x5)3 as:
x5⋅x5⋅x5
To multiply exponents with the same base, simply add the exponents. Therefore, x5⋅x5⋅x5 = x5+5+5 = x15.
There's actually an even shorter way to simplify expressions like this. Take another look at this equation:
(x5)3 = x15
Did you notice that 5⋅3 also equals 15? Remember, multiplication is the same as adding something more than once. That means we can think of 5+5+5, which is what we did earlier, as 5 times 3. Therefore, when you raise a power to a power you can multiply the exponents.
Let's look at one more example:
(x6)4
Since 6⋅4 = 24, (x6)4 = x24
x24
Let's look at one more example:
(3x8)4
First, we can rewrite this as:
3x8⋅3x8⋅3x8⋅3x8
Remember in multiplication, order does not matter. Therefore, we can rewrite this again as:
3⋅3⋅3⋅3⋅x8⋅x8⋅x8⋅x8
Since 3⋅3⋅3⋅3 = 81 and x8⋅x8⋅x8⋅x8 = x32, our answer is:
81x32
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Notice this would have also been the same as 34⋅x32.
Still confused about multiplying, dividing, or raising exponents to a power? Check out the video below to learn a trick for remembering the rules:
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Enter values into any two of the input fields to solve for the third.
What is an exponent?
Exponentiation is a mathematical operation, written as an, involving the base a and an exponent n. In the case where n is a positive integer, exponentiation corresponds to repeated multiplication of the base, n times.
an = a × a × ... × a
n times
The calculator above accepts negative bases, but does not compute imaginary numbers. It also does not accept fractions, but can be used to compute fractional exponents, as long as the exponents are input in their decimal form.
Basic exponent laws and rules
When exponents that share the same base are multiplied, the exponents are added.
an × am = a(n+m)
EX: 22 × 24 = 4 × 16 = 64
22 × 24 = 2(2 + 4) = 26 = 64
When an exponent is negative, the negative sign is removed by reciprocating the base and raising it to the positive exponent.
| a(-n) = |
|
| EX: 2(-3) = 1 ÷ 2 ÷ 2 ÷ 2 | = |
|
| EX: 2(-3) = |
| = |
|
When exponents that share the same base are divided, the exponents are subtracted.
| = a(m - n) |
| EX: |
| = |
| = |
|
| = 2(2-4) = 2-2 = |
| = |
|
When exponents are raised to another exponent, the exponents are multiplied.
(am)n = a(m × n)
EX: (22)4 = 44 = 256
(22)4 = 2(2 × 4) = 28 = 256
When multiplied bases are raised to an exponent, the exponent is distributed to both bases.
(a × b)n = an × bn
EX: (2 × 4)2 = 82 = 64
(2 × 4)2 = 22 × 42 = 4 × 16 = 64
Similarly, when divided bases are raised to an exponent, the exponent is distributed to both bases.
| ( |
| )n | = |
|
| EX: ( |
| )2 | = |
| × |
| = |
|
| ( |
| )2 | = |
| = |
|
When an exponent is 1, the base remains the same.
a1 = a
When an exponent is 0, the result of the exponentiation of any base will always be 1, although some debate surrounds 00 being 1 or undefined. For many applications, defining 00 as 1 is convenient.
a0 = 1
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Shown below is an example of an argument for a0=1 using one of the previously mentioned exponent laws.
If an × am = a(n+m)
Then an × a0 = a(n+0) = an
Thus, the only way for an to remain unchanged by multiplication, and this exponent law to remain true, is for a0 to be 1.
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When an exponent is a fraction where the numerator is 1, the nth root of the base is taken. Shown below is an example with a fractional exponent where the numerator is not 1. It uses both the rule displayed, as well as the rule for multiplying exponents with like bases discussed above. Note that the calculator can calculate fractional exponents, but they must be entered into the calculator in decimal form.
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It is also possible to compute exponents with negative bases. They follow much the same rules as exponents with positive bases. Exponents with negative bases raised to positive integers are equal to their positive counterparts in magnitude, but vary based on sign. If the exponent is an even, positive integer, the values will be equal regardless of a positive or negative base. If the exponent is an odd, positive integer, the result will again have the same magnitude, but will be negative. While the rules for fractional exponents with negative bases are the same, they involve the use of imaginary numbers since it is not possible to take any root of a negative number. An example is provided below for reference, but please note that the calculator provided cannot compute imaginary numbers, and any inputs that result in an imaginary number will return the result 'NAN,' signifying 'not a number.' The numerical solution is essentially the same as the case with a positive base, except that the number must be denoted as imaginary.