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# Radicals and rational exponents — Harder example

Watch Sal work through a harder Radicals and rational exponents problem.

## Want to join the conversation?

• At , he explains that 1/3 is the same as 3^-1. Is this the only way to figure this problem out, or is there another way?
• Yes, you could try approaching the 3^-1/5 differently by changing it to the fraction 1/[3^(1/5)]. Working with fractions is much more difficult than just working with the plain numbers though, so I would just use Sal's method because it saves you time.
• I'll be honest I didn't understand this.
• are there any other videos like this on Khan Academy. because I did not understand how to do this
(1 vote)
• There are videos that can teach you exponent properties, which will make this easier for you.
• At , why don't you get 9 raised to the 1/5 since you are multiplying 3x3?
(1 vote)
• It sort of looks like that, if you squint and ignore the exponents. But 3^ (-1/5) ∙ 3^ (2/5) does not have any 3's in it really. It is `1 divided by the 5th root of 3` times the `5th root of three squared`.

What we have to do instead is use the exponent rules that say that
` xᵃ ∙ xⁿ = xᵃ⁺ⁿ `
and this applies even when a and n are fractions ("rational exponents")

Let's say that we have `9² ∙ 9^½` (just an example that is easier to deal with than the inverse of the 5th root of 3)
We cannot multiply 9 times 9 to get 81 and then do the exponent addition, which would be 2 + 1/2 = 4/2 + 1/2 = 5/2 as the exponent. Then we would have 81^5/2
That number would be 59049

Instead, 9² = 81 and 9^½ = 3, so `9² ∙ 9^½` = 81 ∙ 3 = only 243 and that is a lot less than 59049

When we apply the exponent rules for products of powers of the same number, we don't multiply the base, we just add the exponents

In this math problem, we have exponents of -1/5 + 2/5 which result in 1/5 as the simplified exponent

By the way, that is the fifth root of 3 which is ⁵√3
• I did not understand this and to worry more this will be in my sat tommorow
• You can look for other videos or tell us the part where you didn't get it.
Maybe in your next test try not to leave topics for the very end.
• why at the end you got 16 how ?
• At the end, we are trying to figure out 2^(2x - 3y). We already know from the question that 2x - 3y is 4, so we can substitute this into our first expression:
2^(2x - 3y) = 2^4
This is the same as 2 * 2 * 2 * 2, which is just 16.