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Course: Algebra (all content)>Unit 11

Lesson 3: Rational exponents intro

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.

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• At , 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?"
• 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.
• Should I memorize some of the basic exponents?
Example:
4^3
• I recommend memorizing the perfect squares up to 30 especially if u want to compete in math.
• 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?
• Take the root equivalent to the denominator (bottom), and raise to the power of the numerator (top).
• What do you do when you have for example 2/3 to the power of 2?
• so if 8^1/3 = ³√8 then would 8^2/3 = 2³√8?
• 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

• 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?
• 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!
• 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?
• 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
Another example, 32^(2/5) means take the fifth root of 32, then raise that to the 2nd power (getting 4 as the answer).
• 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?
• 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?
• 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.
• Why is anything raised to the zero power 1? It doesn't make sense to me and seems like a made-up answer. It seems to me like anything raised to the zero power should equal zero. That kind of answer seems more logical.
(1 vote)
• The concept of a number raised to the zero power equals one can be explained in several ways and is based on basic multiplicative concepts. Looking at the pattern established when a number is raised to different powers, each one less than the next, helps explain the concept.

When a number such as 2 is raised to different powers, a particular pattern is seen as the exponent changes:

2^6 = 2*2*2*2*2*2 = 64 2^5 = 2*2*2*2*2 = 32 2^4 = 2*2*2*2 = 16 2^3 = 2*2*2 = 8 2^2 = 2*2 = 4 2^1 = 2

As the exponent value moves from 6 to 1, we see that the resulting values are reduced, consecutively, dividing by 2: 64/2 = 32, 32/2 = 16, 16/2 = 8, 8/2 = 4 and 4/2 = 2. Extrapolating from this pattern, an exponent of 0 will result in an answer of 2/2 = 1, proving 2^0 = 1.

The number 2 was used to provide an example; however, this concept applies to all nonzero numbers.