## Destinyyyy one year ago Can someone explain this to me?

1. Destinyyyy

Simplify the following expression. Express the answer w/positive exponents (3xy^-1/y^3) ^-4

2. Destinyyyy

= (3x/ y^-1 y^3) ^-4 = (3x/y^2) ^-4 (3x/y^2) ^-4 = (3x)^-4/(y^2)^-4 = 3^-4 x^-4/y^-8 = y^8/81x^4

3. Destinyyyy

I know that my answer is wrong.. Where did I mess up?

4. phi

it's not clear what the starting expression is. $\left(\frac{3xy^{-1}}{y^3}\right)^{-4}$?

5. Destinyyyy

Yes thats correct.. Sorry I dont get how to use the equation thing.

6. phi

ok so the y's inside the parens can be handled a few ways. one way: write y^-1 as 1/y $\left(\frac{3x}{y \cdot y^3}\right)^{-4} \\$

7. phi

the other way: y^-1 / y^3 . keep the base y. new exponent is top minus bottom exponents: -1 - 3 = -4. so y^-4 $\left(3xy^{-4}\right)^{-4} \\ =\left(\frac{3x}{y^4}\right)^{-4} \\$

8. phi

I would "flip" the fraction and change the sign of the -4 exponent to +4 $\left(\frac{y^4}{3x}\right)^{4}$

9. phi

can you finish ?

10. Destinyyyy

Um I think so... Give me a second

11. Destinyyyy

I have y^-16/ 3^4 x^-4 = y^-16/?

12. Destinyyyy

I know the final answer is y^16/(3x)^4 but im stuck on the last part

13. phi

if you start with $\left(\frac{y^4}{3x}\right)^{4}$ all the exponents stay positive

14. phi

if we do it "brute force" remember that $\left(\frac{y^4}{3x}\right)^{4} = \left(\frac{y^4}{3x}\right) \left(\frac{y^4}{3x}\right) \left(\frac{y^4}{3x}\right) \left(\frac{y^4}{3x}\right)$

15. phi

and when you multiply fractions, you multiply top times top y^4 * y^4 * y^4 *y^4 = y^16 (the short way is to use the rule $(a^b)^c= a^{b\cdot c}$

16. phi

and the bottom is $\left( 3x\right)^4$

17. phi

or 3^4 * x^4 or 81x^4 there are a few ways to write it.

18. Destinyyyy

Alright thank you!

19. phi

btw, when you started *** Simplify the following expression. Express the answer w/positive exponents (3xy^-1/y^3) ^-4 **** on the next post you say = (3x/ y^-1 y^3) ^-4 = (3x/y^2) ^-4 notice you changed 3xy^-1 to 3x/y^-1 when you do that (move y^-1 from the "top" to the "bottom"), you should change the sign of the exponent. you should have written: = (3x/ y^1 y^3) ^-4 and then you get = (3x/y^4) ^-4 and now you will get the correct answer

20. Destinyyyy

I see

21. phi

and to complete the thought... the other way to do this is $\left(\frac{y^4}{3x}\right)^{4}= \frac{\left( y^4\right)^4}{\left( 3x\right)^4}$ and then simplify the top using the rule $(a^b)^c= a^{bc}$ to get $$y^{4\cdot 4} = y^{16}$$