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- anonymous

Rewrite with rational exponents. Can someone please help me understand and work through this problem?
\[\sqrt[6]{xy^5z}\]

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- anonymous

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- anonymous

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

- anonymous

It says re-write with rational exponents

- anonymous

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

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- myininaya

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

- anonymous

Ok. So here's basically what you need to know to do all of these problems:
\[\huge \sqrt[a]{b^kc^j} = b^{\frac{k}{a}}c^{\frac{j}{a}}\]

- myininaya

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

- anonymous

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?

- anonymous

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

- anonymous

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

- anonymous

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

- anonymous

They are not. Remember the last problem you did?
\[\sqrt[4]{(5a)^4} = (5a)^{\frac{4}{4}} = (5a)^1 = 5a\]

- anonymous

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

- anonymous

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

- anonymous

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

- anonymous

Raise a product to a power that is.

- anonymous

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

- anonymous

What is the exponent on the x ?

- anonymous

1?

- phi

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

- anonymous

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

- anonymous

thank you.

- anonymous

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

- anonymous

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

- anonymous

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

- myininaya

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

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