If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

### Course: Pre-algebra>Unit 5

Lesson 3: Powers of fractions and decimals

# Exponents review

Review the basics of exponents and try some practice problems.

## Exponents and bases

Here's what an exponent and a base look like:
${4}^{3}$
The small number written above and to the right of a number is called an $\text{exponent}$. The number underneath the exponent is called the $\text{base}$. In this example, the base is $4$, and the exponent is $3$.

## Evaluating exponents

An exponent tells us to multiply the base by itself that number of times.
In our example, ${4}^{3}$ tells us to multiply the base of $4$ by itself $3$ times:
$\begin{array}{rl}{4}^{3}& =4×4×4\\ \\ \phantom{{4}^{3}}& =64\end{array}$

### What about when the exponent is a zero?

Any base with an exponent of zero is equal to $1$.
For example, ${7}^{0}=1$.
Check out this video to see why.

## Practice

Problem 1
Evaluate.
${9}^{2}=$

Want to try more problems like this? Check out this exercise.

## Want to join the conversation?

• why is 5 to the 0 power 1?
• Good question! Look at the following exponents:
2^4 = 16
2^3 = 8
2^2 = 4
2^1 = 2
2^0 = __
Respective to the pattern, what comes next?! 1!

Take a look at this:
2^(m-n) = 2^m/2^n
If m = n...
2^(1-1) = 2^1/2^1
2^0 = 1

Hope this helps!
• i get confused and multiply ex. 5 x 2 =10 when its 25 how can i rember to times it by 2 also how is 5 times 0 equal to 1? ':(
• So for exponents you need to think about it a bit different. I will use the example you gave of 5 raised to the 2nd exponent (5^2) for my explanation. The exponent (the number 2) is the number of bases (the number 5) you multiply together. So for 5^2, you would use two 5's and multiply them together which is simply 5x5=25. So for another example if we lower the exponent to 1, we would be looking at 5^1. Well let's apply the same principle of using just one 5, which is simply 5=5.

Let's move on to your second question which is a touch more complicated. I will simplify it for you though. Once your exponent is less than 1 the rules get a little different and you start dealing with fractions. 5^0 = 5*(1/5) = 1. The exponent in this case is the number + 1 that you divide the base number by. I illustrated it with multiplying it by a fraction, but the principle is still the same. I know this can be a difficult topic to understand at first, and explanation isn't the exact proof/theorem, but I do hope it helps you get a basic understanding of exponents.
• how does 5 with the exponent of 0 have the answer of 1.
• Well, I think that CycoCyco answered it somewhat well, but here's another explanation from me:

When having an exponent (such as 5 to the power of 2), you're setting up 5^2, or 5 * 5, which equals one. Same with having five raised to the power of one, which equals five.

In earlier grades, you leaned that 5 * 0 = 0. But in math, 5^0 = 1, because you're not raising the power by anything.

I hope this explanation helped.
• I dont understand why 5'0 pwr is = to 1
• Hi @adam.39594, I'd suggest that you look at the video in Exponents, The Zeroth Power. I'm not sure if we're allowed to put links, so go look at that video... Or search The Zeroth Power in Khan Academy. Hope this helps!
- V
• What about negative exponents? How do you figure those out?
• Because
5^3=125
5^2=25
5^1=5
5^0=1
5^-1=.2
Just divide.
• Idk why the power of 0 is always 1 but it makes it easier for me.
• how is 5 to the 0 power not 0 because 5 zero times is 0
• In general, x to the y power is usually not x times y. So it is a mistake to assume that 5 to the 0 power is 5 times 0.

Look at the following pattern:

5^4 = 5 * 5 * 5 * 5
5^3 = 5 * 5 * 5 = (5 * 5 * 5 * 5)/5
5^2 = 5 * 5 = (5 * 5 * 5)/5
5^1 = 5 = (5 * 5)/5

As we can see, each time the exponent goes down by 1, the answer is divided by 5. Continuing the pattern gives 5^0 = 5/5 = 1.

Have a blessed, wonderful day!
• I don’t get 5 to the power 0
• Any positive value to the power of 0 equals 1. The reason is that, since exponents represent repeated multiplication, then applying the power of 0 is essentially dividing a value by itself the specific number of times. For example, if we have 5^0, then that means we are multiplying it by its reciprocal the specific number of times it got multiplied. So when we have 5^0, are multiplying 5 by itself a specific number of times, then dividing by 5 the same exact number of times it got multiplied. Therefore, every single time a 5 was multiplied, it will be removed by division through the reciprocal, ending up with a 1.
• I keep on getting confused with the base and the exponent. Can someone help?
• The base is the number below for example in 5^2, 5 is the base and 2 is the exponent. Which means that this would expand to 5 x 5 = 25. The exponent tells you how many times the base will be multiplied by itself, we knew that the 5 would be multiplied by itself twice because the exponent was 2.
• how do you add or multiply exponents together without getting rid of the exponents? for example if you had -2x^4+4x^2+5 -7x^5 would you add all the exponents together or would you leave them by their selves?
• You have 4 unlike terms. You can't combine any of them.
Remember, exponents represent repetitive multiplication of a common base. You only change an exponent if you are multiplying or dividing a common base. In your expression, the operation is addition/subtraction. So, the exponents can't be changed.
Hope this helps.