Question

Someone please explain Rational Exponents, (please be descriptive)

Answer 1

\[x^{\frac{a}{b}} = \sqrt[b]{a}\]

Answer 2

\[b \neq 0\]

Answer 3

I want to find out an easy way to simplify working out rational exponents given an exponent within the root of a number (Example: 4^sqrt300 = n). Too many sources show how exponents from roots can be broken into fractional exponents of the rooted number, but what's the work needed to just get the answer????

Answer 4

@JoãoVitorMC That is very very wrong.

Answer 5

@MOJUTMNT You want a method of calculating rational numbers?

Answer 6

Yes. ^~^

Answer 7

yes sorry i missed the "x"
\[x^{\frac{a}{b}} = \sqrt[b]{x^a}\]

Answer 8

Through roots.

Answer 9

Okay, the first thing you should do is simplify the fraction to the simplest form to make your life easy. Then do this: \[
c^{a/b} = (c^a)^{1/b}
\]Now, to make things easier we'll say \(x = c^a\) and use \(x\) from here on out. I suppose it wouldn't be hard for you to find \(x\), right?

Answer 10

We want to find \(x^{1/b}\)

Answer 11

Do you understand how it doesn't really matter what the numerator is, that the hard part is really just in the denominator?

Answer 12

Before you continue, this is done through square roots.

Answer 13

Can you give an example of what you mean?

Answer 14

For instance, I mean like the square root of 300 with an exponent of 4 next to the root.

Answer 15

\[
\sqrt{300^4}
\]?

Answer 16

With the 4 to the left of the square root as an exponent.

Answer 17

\[
\sqrt[4]{300}
\]?

Answer 18

Yes.

Answer 19

Do you know how to factor out a number?

Answer 20

Do you mean by finding the highest number divisible?

Answer 21

Factorizing a number...
Here's an example:

Answer 22

So I have to find the lowest number divisible, and then break it down by repeating that same process?

Answer 23

You want to divide out any prime numbers, until all that is left is a bunch of prime numbers.

Answer 24

This is perfect, can I show you an example of a problem I have?

Answer 25

Where does the exponent of 4 come into play?

Answer 26

There is no exponent of 4, that's multiplication\[135 = 5 *3 *3 *3 = 5* 3^3 \]
so now your \[\large 4 * \sqrt[3]{135} \]
\[\huge 4 * \sqrt[3]{5*3^3}\]

Answer 27

\[\large 4 * \sqrt[3]{5*3^3} = 4 * \sqrt[3]{5}*\sqrt[3]{3^3}\]

Answer 28

No, the four is an exponent... just like the 3 is in the new problem you just created.

Answer 29

I didn't create a problem... i was using your image: http://assets.openstudy.com/updates/attachments/51467a27e4b04cdfc582af15-mojutmnt-1363575901148-rationalexponent.png

Answer 30

My fault. That question was meant for the previous problem.

Answer 31

Before we continue can you show me how the exponent of 4 was used in the previous problem.

Answer 32

maybe a helpful hint, exponents can be written multiple ways

ex. these are equivalent
\[\sqrt[4]{x}=x^{\frac{ 1 }{ 4 }}\]

Answer 33

was that the square root to fraction, you were looking for?

Answer 34

Oops, my bad, 300 = 3*2*2*5*5 NOT 2*2*5*5*5

If you understand up to here, then... \[\large (3*2^2*5^2)^{\frac{ 1 }{ 4 }} = 3^{\frac{ 1 }{4 }} (2^2)^{\frac{ 1 }{ 4 }} * (5^2)^{\frac{ 1 }{ 4}}\]

Answer 35

Thank you very much for that answer. Thank you all for being descriptive. All I needed was for someone to spell it out for me.

Answer 36

@agent0smith is that the final answer?

Answer 37

Not yet. Can you simplify it from there? Note that \[\huge (a^b)^c = a ^{b*c}\] try using it on this:
\[\large 3^{\frac{ 1 }{4 }} *(2^2)^{\frac{ 1 }{ 4 }} * (5^2)^{\frac{ 1 }{ 4}}\]

Answer 38

Yes I can, thank you. Now back to the next question I will post a new screen shot.

Answer 39

This was Q2:\[ \large 4 * \sqrt[3]{5*3^3} = 4 * \sqrt[3]{5}*\sqrt[3]{3^3} \] \[\large 4 * \sqrt[3]{5}*({3^3})^{\frac{ 1 }{3 }}\]