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### Course: Class 10 (Foundation)>Unit 1

Lesson 3: Exponents

# Evaluating fractional exponents

Sal shows how to evaluate 64^(2/3) and (8/27)^(-2/3). Created by Sal Khan.

## Want to join the conversation?

• At , can you explain why the reciprocal eliminates the negative sign? What does flipping the fraction do that makes it a positive?
• I think the pattern goes something like this:
2^3 = (2.2.2) = 8
2^2 = (2.2) = 4
2^1 = (2) = 2
2^0 = ? (This is tricky and is covered in later videos, it is actually = 1, but I'm not yet exactly sure why!)
Then...
2^-1 = (1/2) = 1/2
2^-2 = (1/2.1/2) = 1/4
2^-3 = (1/2.1/2.1/2) = 1/8
So the "negative" exponent just indicates that the base number (2) is now the reciprocal (1/2) - or flipped.
So since 2^-3 is 1/8, and 1/2^3 is also 1/8, then...
To get from 2^-3 to 1/2^3 all you have to do is flip 2/1, it becomes 1/2, and change the sign of the exponent -3 to +3 and both still equal each other, and equal the same answer = 1/8
• So is x^2/3 the same thing as finding the cube root of x, and then squaring it? (or vice versa)
• . He is correct. (27)^2/3=the cube root of 27 and square it=3^2=9
• I'm stuck on a problem.
How would you simplify the following: (x^3)^(2/3)

My first thought would be to multiply the exponents: 3/1 * 2/3 which would leave me with an exponent of 2. Can anyone confirm this answer for me?
• If my brain does not fail me I think that's correct. The answer is x^2.
• well what if something was like 1/2 to the power of 7 how would you
solve that?
• (1/2)^7 is telling you to multiply: 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2
Hope this helps.
• at did Sal mean numerator
• He means "we're doing the same thing down here that we're doing in the numerator."
(1 vote)
• how do you make this solution work when you are working with negatives? Example: (-7776)^(2/5) or 〖4096〗^(-5/6)
• With a negative number inside the root, you cannot take the root if it is even (the denominator of the fraction), but it if it is odd, then the answer will end up negative. 7776 = 6^5 (rather than going through factoring, I did 7776^(1/5) in calculator), so squaring we end up with (-6)^2 which ends up as 36. With a negative exponent, this causes the expression to reciprocate and change exponent to positive, so start with 1/(4096)^(5/6) = 1/4^5 = 1/1024.
• I have a question..
Why is (-9)^1/2 not a real number?
• What number can you multiply by itself to get a negative number? A positive times a positive is positive, but also a negative times a negative is positive.
• At , Sal says that "we already seen how to think about something like 64 to the 1/3 power." Where is the video that teaches that?
• How does one solve a number to a fraction power, such as six to the power of one eleventh? I still do not get it
(1 vote)
• Fractional powers, also called rational exponents, are a different way of writing roots of numbers, the numerator is the power of the term inside the root and the denominator is the power of the root. SO 6^(1/11) would be the same as the eleventh root of 6, written with a six inside the root sign and a small 11 on the crook of the root sign (√) which is sort of inside the V part of the root sign. There is nothing to solve unless you want an approximation which you can get by entering 6^(1/11) into a calculator.