Question

Rewrite with rational exponents. Can someone please help me understand and work through this problem?

\[\sqrt[6]{xy^5z}\]

Answer 1

Ok well do they want you to re-write it with fractional exponents? or ?

Answer 2

It says re-write with rational exponents

Answer 3

Okay, well sqrt6 is like a power of 1/6. Let's see if we can apply that..

Answer 4

\[\sqrt[a]{x}=x^\frac{1}{a}\]

Answer 5

Ok. So here's basically what you need to know to do all of these problems:

\[\huge \sqrt[a]{b^kc^j} = b^{\frac{k}{a}}c^{\frac{j}{a}}\]

Answer 6

\[\sqrt[a]{b^kc^n}=(b^k c^n)^\frac{1}{a}=b^\frac{k}{a}c^\frac{n}{a}\]

Answer 7

so how do I start so I take the b and move it over? make a the denominator and make k and j the numerators?

Answer 8

The index of your radical what you will divide your powers by.

Answer 9

There's an 'is' missing from that sentence.

Answer 10

I am still confused. I don't know where to begin. I feel like all of these problems are SOOO different

Answer 11

They are not. Remember the last problem you did?

\[\sqrt[4]{(5a)^4} = (5a)^{\frac{4}{4}} = (5a)^1 = 5a\]

Answer 12

It is easy to you because you understand I don't. I just don't learn that way. I need to see how it is solved. I have to actually be able to do the problem

Answer 13

So in this case we have:
\[\sqrt[6]{xy^5z}\]
We can rewrite it as:
\[\large (xy^5z)^{\frac{1}{6}}\]

Answer 14

And the recall that when you raise a power to a power you multiply the exponents.

Answer 15

Raise a product to a power that is.

Answer 16

You just divide each of the exponents by the index of the radical.

Answer 17

What is the exponent on the x ?

Answer 18

1?

Answer 19

Let's start at the beginning.
Do you know how to write
\[\sqrt{x}= x^{?}\]

Answer 20

Correct. So now divide that 1 by 6 and the new exponent on the x will be 1/6

Answer 21

thank you.

Answer 22

Now do the same thing for the y. The exponent on the y is 5, so 5 divided by 6 is 5/6

Answer 23

Then again for the z and your result is:
\[\large x^{\frac{1}{6}}y^{\frac{5}{6}}z^{\frac{1}{6}}\]

Answer 24

which doesn't seem 'simpler' at all. But that's sometimes how that goes.

Answer 25

another general example of applying law of exponent:
\[(x^nyz^m)^{r}=(x^ny^1z^m)^r=x^{nr}y^{1r}z^{mr}=x^{nr}y^rz^{mr}\]