## mathtard 4 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: $(x^nyz^m)^{r}=(x^ny^1z^m)^r=x^{nr}y^{1r}z^{mr}=x^{nr}y^rz^{mr}$
{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "url": "http://openstudy.com/updates/4e3c715b0b8bfc76a3f67d60", "fetch_time": 1448774108000000000, "content_mime_type": "text/html", "warc_filename": "crawl-data/CC-MAIN-2015-48/segments/1448398456289.53/warc/CC-MAIN-20151124205416-00010-ip-10-71-132-137.ec2.internal.warc.gz", "warc_record_offset": 167486497, "warc_record_length": 12163, "token_count": 757, "char_count": 2263, "score": 4.15625, "int_score": 4, "crawl": "CC-MAIN-2015-48", "snapshot_type": "longest", "language": "en", "language_score": 0.8524670004844666, "file_path": "/sailhome/yjruan/project/latent-lm/data/processed/finemath-4plus/finemath-4plus/train-00018-of-00064.parquet"}