Main content

## Get ready for AP® Calculus

# Intro to rational exponents

What does it mean to take a number by a power which is a unit fraction? For example, what is the result of 3 raised to ½? Created by Sal Khan.

## Want to join the conversation?

- At1:20, Sal asks, "But what is the square root of 4, especially the principle root mean?" and then goes on to ask, "what is a number that if I were to multiply it by itself...I'm going to get 4." I understand that +2 is apparently "the principle root", but what about -2, which if multiplied by itself is also equal to 4? What kind of root is -2 and why is it that only the "principle root" is given as the answer to "what is the square root of 4?"(140 votes)
- The principle root and square root are two different things. The square root is asking that question, which number squared equals that number, say, 4. This does leave two answers, positive and negative, so
**you were correct**. However, the principle root is basically the absolute of the square root, or √|x|, which means that it is only positive. This was created, I think, for geometry, because you can't have a triangle with side lengths of -3, -4, and -5. Also, if this is a comfort to you, I didn't know about principle roots until recently. We'd say that the square root of 4 is ±2, for that same reason you mentioned.(140 votes)

- Should I memorize some of the basic exponents?

Example:

4^3(24 votes)- I recommend memorizing the perfect squares up to 30 especially if u want to compete in math.(9 votes)

- In fractional exponents, i'm curious on what to do if there is a fraction such as 5/7 or 9/17 as an exponent. Do you take the square root and then multiply, or do something else?(22 votes)
- Take the root equivalent to the denominator (bottom), and raise to the power of the numerator (top).(27 votes)

- What do you do when you have for example 2/3 to the power of 2?(13 votes)
- Petra, did you mean 2 to the power of 2/3? If so, that is

2 times the cube root of 2.(0 votes)

- so if 8^1/3 = ³√8 then would 8^2/3 = 2³√8?(4 votes)
- No, unfortunately this would be wrong. You propably have not learned this yet, but you can rewrite any exponential expression of the form x^(n*m) as (x^n)^m.

So when you look at your example of 8^2/3, you could rewrite it as 8^(2*1/3).

By matching the corresponding parts to x^(n*m), this could then be expressed in the form of (x^n)^m:

(8^2)^(1/3)

= 64^(1/3)

= 4

Alternatively you could swap the 2 and the 1/3, which might make the problem easier. You can do this because of the the commutative property of multiplication, which allows you to "choose" wether you see 2 as the m or the n (the same thing goes for the 1/3). This would give you:

(8^(1/3))^2

= (2)^2

= 4

I hope this will help you.(21 votes)

- At no part of the video did Sal explain what to do if a number is raised to, for example, the 2/5th power? What if the fraction has an integer larger than 1 for its numerator?(5 votes)
- What a good question!

If you have ANY fractional power, the denominator tells what root to take and the numerator tells what power to raise that number to.

For example, 16^3/2 means take the square root of 16, then raise that to the 3rd power

(getting 64 as the answer).

Another example, 32^(2/5) means take the fifth root of 32, then raise that to the 2nd power (getting 4 as the answer).(5 votes)

- I know that 7^0 is one, and is so for all numbers other than zero. But what I want to know is why it isn't zero. Or, in other words, why isn't 7^0 equal to zero?(6 votes)
- Watch these 2 videos as Khan explained it. Maybe it would help clarify it.

0 and 1st Power

https://www.khanacademy.org/math/pre-algebra/exponents-radicals/world-of-exponents/v/raising-a-number-to-the-0th-and-1st-power

Powers of 0

https://www.khanacademy.org/math/pre-algebra/exponents-radicals/world-of-exponents/v/powers-of-zero(5 votes)

- okay what are the 3 cube roots of 8? (cube numbers have 3 roots, square numbers have 2 roots)

by cube number I mean x^3=y

by square number I mean x^2=y

now I already know 2 is a cube root but it is not the Only cube root. there are two others. what are those other cube roots?(6 votes)- The other cube roots are 2 and 2. 8 is just 2*2*2. Or 2^3. That is it's prime factorization, nothing else. Hope this helped!(2 votes)

- What if I had an improper fraction?

For example: 25 to the power of 3/2(3 votes)- It would be equivalent to sqrt(25)^3, which would be 5^3 = 125.(3 votes)

- Sal writes all the rational exponents as
**fractions**. But can exponents be in**decimal**form?

For example,*x^(-2.5)*and*x^(-5/2)*, are both of them correct?(2 votes)- The two versions are equivalent. However, the fraction form is easier to understand. The denominator of the fraction tells you the radical index. You have a denominator of 2, so it indicates a square root. If the denominato is 3, then the problem is working with a cube root. It it is 4, then the problem is working with a 4throot.

If you have a problem like: (-8)^(2/3) you can see that you need to do a cube root (the 3 in the denominator) and then square the result (the 2 in the numerator.

(-8)^(2/3) = cubert(-8)^2 = (-2)^2 = 4

If the exponent is in decimal form, that information is not visible. You would have to convert to a fraction to make the info visible. There is also the risk that you convert a fraction to decimal, find it repeats and you then round the decimal value. If you happen to do this, then you have changed the exponent. For example: An exponent of 1/3 = Do a cube root. If you convert it to decimal form: 1/3 = 0.33333... with the 3 repeating. If it gets rounded to 0.3, the exponent would then be 3/10 which means do the 10th root, then cube the result.

Hope this helps.(5 votes)

## Video transcript

We already know a good
bit about exponents. For example, we know
if we took the number 4 and raised it to
the third power, this is equivalent
to taking three fours and multiplying them. Or you can also view it
as starting with a 1, and then multiplying the 1
by 4, or multiplying that by 4, three times. But either way, this is
going to result in 4 times 4 is 16, times 4 is 64. We also know a little bit
about negative exponents. So for example, if I were take
4 to the negative 3 power, we know this negative
tells us to take the reciprocal 1/4 to the third. And we already know
4 to the third is 64, so this is going to be 1/64. Now let's think about
fractional exponents. So we're going to think about
what is 4 to the 1/2 power. And I encourage you
to pause the video and at least take a guess
about what you think this is. So, the mathematical
convention here, the mathematical definition
that most people use, or in fact that all people use here,
is that 4 to the 1/2 power is the exact same thing
as the square root of 4. And we'll talk in the
future about why this is, and the reason why this
is defined this way, is it has all sorts of
neat and elegant properties when you start manipulating
the actual exponents. But what is the
square root of 4, especially the
principal root, mean? Well that means,
well, what is a number that if I were to
multiply it by itself, or if I were to have
two of those numbers and I were to multiply
them, times each other, that same number,
I'm going to get 4? Well, what times
itself is equal to 4? Well that's of
course equal to 2. And just to get a sense of why
this starts to work out, well remember, we could
have also written that 4 is equal to 2 squared. So you're starting to see
something interesting. 4 to the 1/2 is equal to
2, 2 squared is equal to 4. So let's get a couple
more examples of this, just so you make sure
you get what's going on. And I encourage you to pause
it as much as necessary and try to figure
it out yourself. So based on what
I just told you, what do you think 9 to the
1/2 power is going to be? Well, that's just
the square root of 9. The principal root of 9, that's
just going to be equal to 3. And likewise, we
could've also said that 3 squared is, or
let me write it this way, that 9 is equal to 3 squared. These are both true statements. Let's do one more like this. What is 25 to the
1/2 going to be? Well, this is just
going to be 5. 5 times 5 is 25. Or you could say, 25
is equal to 5 squared. Now, let's think about what
happens when you take something to the 1/3 power. So let's imagine taking
8 to the 1/3 power. So the definition here
is that taking something to the 1/3 power
is the same thing as taking the cube
root of that number. And the cube root is just
saying, well what number, if I had three of that
number, and I multiply them, that I'm going to get 8. So something, times something,
times something, is 8. Well, we already know that 8 is
equal to 2 to the third power. So the cube root of
8, or 8 to the 1/3, is just going to be equal to 2. This says hey,
give me the number that if I say that number, times
that number, times that number, I'm going to get 8. Well, that number is 2 because
2 to the third power is 8. Do a few more examples of that. What is 64 to the 1/3 power? Well, we already know that
4 times 4 times 4 is 64. So this is going to be 4. And we already wrote over here
that 64 is the same thing as 4 to the third. I think you're starting to see
a little bit of a pattern here, a little bit of symmetry here. And we can extend this idea to
arbitrary rational exponents. So what happens if I were
to raise-- let's say I had, let me think of a good number
here-- so let's say I have 32. I have the number 32, and I
raise it to the 1/5 power. So this says hey,
give me the number that if I were to
multiply that number, or I were to repeatedly
multiply that number five times, what is that, I would get 32. Well, 32 is the same thing as
2 times 2 times 2 times 2 times 2. So 2 is that number, that
if I were to multiply it five times, then
I'm going to get 32. So this right over here
is 2, or another way of saying this kind of same
statement about the world is that 32 is equal to
2 to the fifth power.