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## Algebra basics

### Course: Algebra basics > Unit 6

Lesson 2: Negative exponents# Negative exponents review

CCSS.Math:

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:

*Want to learn more about this definition? Check out this video.*

### 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

## Some intuition

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

### Justification #1: Patterns

n | 2, start superscript, n, end superscript |
---|---|

3 | 2, cubed, equals, 8 |

2 | 2, squared, equals, 4 |

1 | 2, start superscript, 1, end superscript, equals, 2 |

0 | 2, start superscript, 0, end superscript, equals, 1 |

minus, 1 | 2, start superscript, minus, 1, end superscript, equals, start fraction, 1, divided by, 2, end fraction |

minus, 2 | 2, 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...

We also know that

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...

And indeed, according to the definition...

## Want to join the conversation?

- how can you say that 1/1/9 is 9??(13 votes)
- 1/(1/9). How many 9ths are in one whole? Nine.(43 votes)

- how do we divide exponents by exponents?(9 votes)
- you subtract the exponent on the top from the exponent on the bottom.(18 votes)

- can we ever go back to just numbers like 2+2(9 votes)
- Simple solution: Invent a time machine.(11 votes)

- What happens when zero is put to the zero power, for example 0^0(4 votes)
- 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!(18 votes)

- 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(9 votes)
- Look at the hints. The hints are very helpful for exponents.(3 votes)

- 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(8 votes)
- In order to fully understand, I found its helpful to draw a number line. On that line would be zero through +8. Then plot or mark where on the line the numbers in the intuitive explanation are. This shows that the neg exponent is doing the opposite of the positive. The number line shows that the positive exponents are high numbers, because the base is increasing exponentially - like multiplication. While the neg exponents are smaller, because the base is exponentially decreasing in size - like division. And it gets more and more small the higher the neg exponent. But it'll never reach or pass zero - which is essentially Zeno's Paradox. I highly recommend checking that out! Understanding Zenos Paradox also helped me understand the depth on neg exponents.

In math, almost everything has an inverse it seems.

Pos numbers vs neg numbers,

Pos exponents vs neg exponents,

Mult vs division.

Hope this helps!(1 vote)

- if a exponent is negative what happens to the base(3 votes)
- 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.(8 votes)

- when you do the opposite of fractions with a negative exponent, do you always have to do reciprocal;(0 votes)
- 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.(8 votes)

- is there any videos on scientific notation in math?(0 votes)
- Here's a list of videos with examples of scientific notation:

https://www.khanacademy.org/math/pre-algebra/pre-algebra-exponents-radicals#pre-algebra-scientific-notation

You can find videos of calculations with scientific notation here:

https://www.khanacademy.org/math/pre-algebra/pre-algebra-exponents-radicals#pre-algebra-computing-scientific-notation

You can also search for anything you need using the search bar at the top left of the site.(7 votes)

- how do i know if the answer will be a fraction or not?(2 votes)
- if it is a fraction you make it in to the whole number for example 1/5^4 = 5^4 is ^ means exponent(1 vote)