## anonymous 4 years ago I am sure there is someone here that can easily tackle this problem! Factor: 3x^(-1/2)+9x^91/2)-81^(-3/2) I panic when I see negative fractional exponents! And my factoring skills are weak at best If you can explain any of the steps or the way you think about the problem I would greatly appreciate the advice.

1. anonymous

$3x^{\frac{-1}{2}}+9x^{\frac{1}{2}}-81^{\frac{-3}{2}}$ like this?

2. anonymous

yes

3. anonymous

I was thinking that an x^(1/2) If 81x^(-3/2) could be rewritten as 81x(-3(1/2)) and 3x(-1/2) could be rewritten as 3x^(-1(1/2))

4. anonymous

i can write out what this means without negative exponents, but i cannot see how to factor it

5. anonymous

oops that an x^(1/2) could be factored out *

6. anonymous

oh wait, is there an x in the last term?

7. anonymous

yes there was

8. anonymous

$3x^{\frac{-1}{2}}+9x^{\frac{1}{2}}-81x^{\frac{-3}{2}}$

9. anonymous

oh yikes I overlooked that

10. anonymous

good catch. I looked over it so quickly the first time I missed the omission.

11. anonymous

ok that is a whole different story. first off you can factor a 3 out of each term, then also perhaps an $x^{\frac{1}{2}}$ if you like. lets try it

12. anonymous

Great I get coefficients of 1 3 and 27

13. anonymous

$3x^{\frac{1}{2}}(x^{-1}+3+27x^{-2})$

14. anonymous

now i wonder if we can go further...

15. anonymous

cool that exactly where I am at the moment as well

16. anonymous

17. anonymous

no i think it is $27x^{-2}$

18. anonymous

because $\frac{-3}{2}-\frac{1}{2}=-2$

19. anonymous

i cannot see how to factor the second part, so i think you are done at that step

20. anonymous

ok so the exponents are not being multiplied in this case

21. anonymous

I was thinking it was -27x^(-3(1/2))

22. anonymous

sorry i was off by a minus sign, should be $-27x^{-2}$

23. anonymous

when you multiply you add the exponents

24. anonymous

so $3x^{\frac{1}{2}}\times -27x^{-2}=-81x^{\frac{1}{2}+(-2)}=-81x^{-\frac{3}{2}}$ which is what you want

25. anonymous

ok I think I see how that works so x^(1/2)x^-2(2/2) = x^(1/2)x^(-4/4) or x^(-3/2)

26. anonymous

Thank you so much for your help! This really helped clarify the concept for me.

27. anonymous

you don't need to change the denominator is 2

28. anonymous

yw

29. anonymous

oops meant (-2(2/2) = (-4/2)

30. anonymous

got it

31. anonymous

awesome!