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.

## Pre-algebra

### Course: Pre-algebra>Unit 11

Lesson 3: Negative exponents

# Negative exponents

Learn how to rewrite expressions with negative exponents as fractions with positive exponents. A positive exponent tells us how many times to multiply a base number, and a negative exponent tells us how many times to divide a base number. We can rewrite negative exponents like x⁻ⁿ as 1 / xⁿ. For example, 2⁻⁴ = 1 / (2⁴) = 1/16. Created by Sal Khan.

## Want to join the conversation?

• I'm confused. If for example 2^4 is 2*2*2*2=16, why is 2^-4 meaning 2/2/2/2 equal to 1/4 rather than 1/16?
• Negative exponents move the value to the other side of a division sign, so 2^-4/1 makes it 1/2^4. Exponents are a shortcut for multiplication, not division.
• i'm confused. so at he says that 1/25/64 is just going to be 64/25 but never explained why? How did he reach that conclusion? i'm so lost!
• I assume it is 1/(25/64), and to divide fractions, you reciprocate (flip) the one in the denominator and multiply, so 1 * 64/25 = 64/25. If it were (1/25)/64, then that would be a different answer 1/25 * 1/64.
• I think a negative exponent is basically the reciprocal of the positive reciprocal. Is this right?
• Slightly yes but better understand that if the power is minus it has to change its place from nominator to denominator or denominator to nominator
AND when it changes its place the minus become positive
• An exponent says how many times to use the base in multiplication. So for example, 2^2 = 2 x 2 = 4.

3^5 = 3 x 3 x 3 x 3 x 3 = 243

Intuitively thinking based on the above: 2^-2:

How does a negative exponent become a reciprocal? That doesn't make sense to me yet.
• It's based on exponent rules. 3^2 x 3^3 would be (3 x 3) x (3 x 3 x 3), or 3^5. So for multiplication of two exponents with the same bases, you add the exponents. What about division? 3^3 / 3^2 is (3 x 3 x 3) / (3 x 3), so it would be 3/1, or 3, which is 3^1. So for division with the same base, you subtract the exponent. If you have 3^3 / 3^3, you would have 3^(3-3) = 3^0 because of this rule, so 3^0 = 3^3/3^3, which turns out to be 1. Anything to the 0th power is 1. if you take 3^0 / 3^1, you have 3^-1, which is also 1/3, so it's the reciprocal. I hope this makes sense to you.
• It's just to clarify that there is a 1. Say we have `3^2 = 9`; `3^1 = 3`; `3^0 = ?`. What would 3^0 be? We know it's 1 and since there are no 3's to multiply 1 with, then we say it's 1. Once you understand the concept, you don't need to write it at all!
• "1 over 25/64 is just going to be 64/25".

Why?

Please explain this in detail (or provide a link to a lesson on this). I do not understand.
• To solve for "1 over 25/64" order of operation says to divide 25/64 first, to get 0.390625. Then, you divide 1/0.390625, you get 2.56. If you try to divide 64/25, you will see that it equals 2.56. In other words: "1 over 25/64" is equal to 64/25. Hope this helps :)
• Is there any other way to understand 2 to the power of -4 and, what this negative symbol does?
• The negative sign on an exponent means the reciprocal. Think of it this way: just as a positive exponent means repeated multiplication by the base, a negative exponent means repeated division by the base.

So 2^(-4) = 1/(2^4) = 1/(2*2*2*2) = 1/16. The answer is 1/16.

Have a blessed, wonderful New Year!
• Couldn't you just do two to the fourth power (which equals 16), and make it into 1/16?
• For the simple reason that 16 is not equal to 1/16. Two to the negative fourth power is 1/16 though.
• I have a question on , why do we put the parentheses around a negative base? If anyone answers this, thank you!