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### Course: Class 8 (Old) > Unit 10

Lesson 2: Negative exponents# Negative exponents

Negative exponents can be rewritten in two ways. Firstly, start with 1 and divide it by 2 the same number of times as the exponent. Secondly, take the reciprocal of the base and raise it to the positive exponent. Created by Sal Khan.

## Want to join the conversation?

- i'm confused. so at6:19he 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!(43 votes)
- 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.(49 votes)

- 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?(38 votes)
- 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.(31 votes)

- 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.(9 votes)- 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.(11 votes)

- why we must start with "1"?(11 votes)
- 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!(7 votes)

- "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.(6 votes)- 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 :)(11 votes)

- I think a negative exponent is basically the reciprocal of the positive reciprocal. Is this right?(8 votes)
- 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(12 votes)

- Is there any other way to understand 2 to the power of -4 and, what this negative symbol does?(4 votes)
- 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!(14 votes)

- I've always been so confused on negative exponents, this helped me a lot!

But I don't understand how 1/(25/64)=64/25?(6 votes)- I asked chatgpt about it and this is the answer it returned. This helped me a lot!

(this is just the excerpt that helped me)

Now, let's consider a fraction like a/b, where "a" is the numerator and "b" is the denominator. Dividing by this fraction means we are asking how many times a fraction with value (a/b) fits into the dividend.

Let's use the dividend "1" for our example: 1 / (a/b).

To determine how many times the fraction (a/b) fits into 1, we can rephrase the question as "what number, when multiplied by (a/b), gives us 1?" In other words, we want to find the multiplicative inverse of (a/b) that, when multiplied, yields 1.

The multiplicative inverse of a fraction a/b is its reciprocal, which is b/a. Why? Because when you multiply a fraction by its reciprocal, the result is always 1.(8 votes)

- Couldn't you just do two to the fourth power (which equals 16), and make it into 1/16?(7 votes)
- For the simple reason that 16 is not equal to 1/16. Two to the negative fourth power is 1/16 though.(2 votes)

- amazing explanation. I wish all teachers would show this rather than just playing the memorization game(5 votes)
- You are 100% right(1 vote)

## Video transcript

We already know that
2 to the fourth power can be viewed as
starting with a 1 and then multiplying
it by 2 four times. So let me do that. So times 2, times
2, times 2, times 2. And that will give us, let's
see, 2 times 2 is 4, 8, 16. So that will give us 16. Now I will ask you a more
interesting question. What do you think 2 to
the negative 4 power is? And I encourage you to pause
the video and think about that. Well, you might be
tempted to say, oh maybe it's negative 16 or
something like that, but remember what the exponent
operation is trying to do. One way of viewing it
is this is telling us how many times are we going to
multiply 2 times negative 1? But here we're going to
multiply negative 4 times. Well, what does negative
traditionally mean? Negative traditionally
means the opposite. So here this is how many times
you're going to multiply. Maybe when we make it
negative this says, how many times are
we going to, starting with the 1, how many times
are we going to divide by 2? So let's think about
that a little that. So this could be
viewed as 1 times, and we're going to
divide by 2 four times. Well, dividing by 2 is the same
thing as multiplying by 1/2. So we could say that
this is 1 times 1/2, times-- let me just
do it in one color. So 1 times 1/2, times
1/2, times 1/2, times 1/2. Notice multiplying
by 1/2 four times is the exact same thing as
dividing by 2 four times. And in this situation
this would get you, well 1/2, well 1 times 1/2 half
is just 1/2, times 1/2 is 1/4, times 1/2 is 1/8,
times 1/2 is 1/16. And so you probably see
the relationship here. If you're-- this is essentially
you're starting with the 1 and you're dividing
by 2 four times. You could also say that 2-- I'm
going to do the same colors-- 2 to the negative 4 is
the same thing as 1/2 to the fourth power. Let me color code
it nicely so you realize what the
negative is doing. So this negative
right over here-- let me do that in
a better color, I'll do it in magenta,
something that jumps out. So this negative
right over here, this is what's causing
us to go one over. So 2 to the negative
4 is the same thing, based on the way we've
defined it just up right here, as reciprocal of 2 to
the fourth, or 1 over 2 to the fourth. And so you could view this as
being 1/2 times 2 times 2 times 2, if you just view
2 to the fourth as taking four 2's
and multiplying them. Or if you use this
idea right over here, you could view it
as starting with a 1 and multiplying it
by 2 four times. Either way, you are
going to get 1/16. So let's do a few
more examples of this just so that we make sure
things are clear to us. So let's try 3 to the
negative third power. So remember, whenever
you see that negative, what my brain
always does is say I need to take the
reciprocal here. So this is going
to be equal to, I'm going to highlight
the negative again, this is going to be 1
over 3 to the third power. Which would be equal to 1/3
times 3 times 3, or 1 times 3 times 3 times 3,
is going to be 27. So this is going to be 1/27. Let's try another example,
I'll do two or three more. So let's take a negative number
to a negative exponent, just to see if we can
confuse ourselves. So let's take the
number negative 4, and let's take it-- I don't want
my numbers to get too big too fast. So let's just take
negative 2 and let's take it to the negative 3 power. I'll make my negatives in
magenta, negative 3 power. So at first this
might be daunting, do the negatives cancel? And that will just be the
remnants in your brain that are trying to think
of multiplying negatives. Do not apply that here. Remember, you see a
negative exponent, that just means the reciprocal
of the positive exponent. So 1 over negative 2
to the third power, to the positive third power. And this is equal to 1
over negative 2 times negative 2 times negative 2. Or you could view it as
1 times negative 2 times negative 2 times
negative 2, which is going to give you 1 over
negative 8 or negative 1/8. Let me scroll over
a little bit, I don't want to have to
start squishing things. So this is equal
to negative 1/8. Let's do one more example,
just in an attempt to confuse ourselves. Let's take 5/8 and raise
this to the negative 2 power. So once again, this negative,
oh I got at a fraction is a negative here. Remember this just means 1
over 5/8 to the second power. So this is just going
to be the same thing as 1 over 5/8 squared,
which is going to be the same thing-- so this
is going to be equal to-- I'm trying to color code it, 1 over
5/8 times 5/8, which is 25/64. 1 over 25/64 is just
going to be 64/25. So another way to
think about it is, you're going to take
the reciprocal of this and raise it to the
positive exponent. So another way you could
have thought about this is 5/8 to the negative 2 power. Let me just take the
reciprocal of this, 8/5 and raise it to the
positive 2 power. So all of these
statements are equivalent. And that would have
applied even when you're dealing with
non-fractions as your base right over here. So 2, you could say well this
is going to be the same thing. 2 to the negative
4 is going to be the same thing as
taking my reciprocal. So this is going to
be the same thing as taking the reciprocal
of 2, which is 1/2 and raising it to
the positive 4 power.