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.

### Unit 1: Lesson 7

Negative exponents

# Negative exponents review

Review the basics of negative exponents and try some practice problems.

## Definition for negative exponents

We define a negative power as the multiplicative inverse of the base raised to the positive opposite of the power:
x, start superscript, minus, n, end superscript, equals, start fraction, 1, divided by, x, start superscript, n, end superscript, end fraction

### Examples

• 3, start superscript, minus, 5, end superscript, equals, start fraction, 1, divided by, 3, start superscript, 5, end superscript, end fraction
• start fraction, 1, divided by, 2, start superscript, 8, end superscript, end fraction, equals, 2, start superscript, minus, 8, end superscript
• y, start superscript, minus, 2, end superscript, equals, start fraction, 1, divided by, y, squared, end fraction
• left parenthesis, start fraction, 8, divided by, 6, end fraction, right parenthesis, start superscript, minus, 3, end superscript, equals, left parenthesis, start fraction, 6, divided by, 8, end fraction, right parenthesis, cubed

### Practice

Problem 1
Select the equivalent expression.
4, start superscript, minus, 3, end superscript, equals, question mark

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

## Some intuition

So why do we define negative exponents this way? Here are a couple of justifications:

### Justification #1: Patterns

n2, start superscript, n, end superscript
32, cubed, equals, 8
22, squared, equals, 4
12, start superscript, 1, end superscript, equals, 2
02, start superscript, 0, end superscript, equals, 1
minus, 12, start superscript, minus, 1, end superscript, equals, start fraction, 1, divided by, 2, end fraction
minus, 22, start superscript, minus, 2, end superscript, equals, start fraction, 1, divided by, 4, end fraction
Notice how 2, start superscript, n, end superscript is divided by 2 each time we reduce n. This pattern continues even when n is zero or negative.

### Justification #2: Exponent properties

Recall that start fraction, x, start superscript, n, end superscript, divided by, x, start superscript, m, end superscript, end fraction, equals, x, start superscript, n, minus, m, end superscript. So...
\begin{aligned} \dfrac{2^2}{2^3}&=2^{2-3} \\\\ &=2^{-1} \end{aligned}
We also know that
\begin{aligned} \dfrac{2^2}{2^3}&=\dfrac{\cancel 2\cdot\cancel 2}{\cancel 2\cdot\cancel 2\cdot 2} \\\\ &=\dfrac12 \end{aligned}
And so we get 2, start superscript, minus, 1, end superscript, equals, start fraction, 1, divided by, 2, end fraction.
Also, recall that x, start superscript, n, end superscript, dot, x, start superscript, m, end superscript, equals, x, start superscript, n, plus, m, end superscript. So...
\begin{aligned} 2^2\cdot 2^{-2}&=2^{2+(-2)} \\\\ &=2^0 \\\\ &=1 \end{aligned}
And indeed, according to the definition...
\begin{aligned} 2^2\cdot 2^{-2}&=2^2\cdot\dfrac{1}{2^2} \\\\ &=\dfrac{2^2}{2^2} \\\\ &=1 \end{aligned}

## Want to join the conversation?

• how can you say that 1/1/9 is 9??
• 1/1/9 = 1*9/1 (We find the reciprocal and multiply) = 9
• what do you do if the question is not 2 x 2 and its something like 2 x 3 then what number to you put down because if its the same number you can just get rid of one of them but if there not what do you do
• Look at the hints. The hints are very helpful for exponents.
• can we ever go back to just numbers like 2+2
• Simple solution: Invent a time machine.
• I do not understand why it becomes a fraction. The intuition does not help either. I know it becomes a fraction, i know the right answer i just do not understand it. For example: 2^-4 i do not get why it becomes 1/2^4
• Sometimes it is helpful for some people to think of it as actual numbers. like if a=5

5 divided by 5 equals 1 = 5^0
1 divided by 5 equals 1/5 =5^-1
1/5 divided by 5 equals 1/5^2
...
• if a exponent is negative what happens to the base
• The base remains the same. As the page explains, a negative exponent just means "the multiplicative inverse of the base raised to the positive opposite of the power". So a^(-b) = 1/(a^b). The base, a, doesn't change. Only its place in the expression changes.
• What happens when zero is put to the zero power, for example 0^0
• Interesting question! Consider the following two rules:

1) Any nonzero number to the zero power is 1.

2) Zero to any positive power is 0.

If we try to extend both rules to define 0^0, we get different answers. So should 0^0 be 0, 1, or something else? Because of this situation, it is best to call 0^0 indeterminate (though 0^0 is often interpreted as 1).

Have a blessed, wonderful day!
• when you do the opposite of fractions with a negative exponent, do you always have to do reciprocal;
• It's not opposite of fractions actually, it's an equivalent expression. they have the equal sign between them. and their value is the same. You can even use the calculator to make sure of that.

4^-2 = (1/4)^2
they are equal, when, and ONLY when:
1- it's a reciprocal.
2- exponent with the opposite sign.

Now let's imagine NOT doing the reciprocal like you said:
4^-2 = 4^2

What's that supposed to be?? Well.. 4^2 is actually the reciprocal!
You can make sure of that by multiplying them (use calculator). You should get 1 if they're truly reciprocals.
4^-2 * 4^2 = 1
Yep!

So keep in mind that to get the number as it is (not changing its value) you have to BOTH flip the fraction (the reciprocal) and flip the sign of the exponent.
• is there any videos on scientific notation in math?