## mathtard 3 years ago Rewrite with rational exponents. Can someone please help me understand and work through this problem? $\sqrt[6]{xy^5z}$

1. polpak

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

2. mathtard

It says re-write with rational exponents

3. simplephysics

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

4. myininaya

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

5. polpak

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

6. myininaya

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

7. mathtard

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?

8. polpak

9. polpak

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

10. mathtard

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

11. polpak

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

12. mathtard

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

13. polpak

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

14. polpak

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

15. polpak

Raise a product to a power that is.

16. polpak

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

17. polpak

What is the exponent on the x ?

18. mathtard

1?

19. phi

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

20. polpak

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

21. mathtard

thank you.

22. polpak

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

23. polpak

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

24. polpak

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

25. myininaya

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