## anonymous 4 years ago how do i simplify nested roots and powers of a single positive variable?

1. TuringTest

$\large (x^a)^b=x^{ab}$

2. TuringTest

$\sqrt[b]{x^a}=x^{a/b}$so those can be used together

3. anonymous

how would i solve this type of problem then?|dw:1328571176207:dw|

4. anonymous

raise the equation to the power if 6 then simplify, then take it to the 1/6

5. anonymous

how'd u get that... if you don't mind.

6. anonymous

got rid of the radicals, using the power rules. easily all you have to do is take the product of each of the nested radicals. make sure you dont forget to apply the power non nested equation. this was simple equation when you get (x +2 + (x)^1/2))^1/2 it becomes very difficult

7. anonymous

|dw:1328571786972:dw| lol. is this what you do though? I know i got a different answer...

8. TuringTest

I can't see the whole problem, is it an equation or simplification?

9. anonymous

simplify

10. TuringTest

$\sqrt{x\sqrt[3]{x^2}}$is all I can see, is there more?

11. anonymous

no lol, that was it

12. anonymous

use the first power rule given by turing test

13. anonymous

i dont understand why i would use the first one and not the second

14. TuringTest

$\large\sqrt{x\sqrt[3]{x^2}}=(x\cdot x^{3/2})^{1/2}=x^{1/2}\cdot x^{(3/2)\cdot(1/2)}$simplify the fractions

15. anonymous

same principal different ways of represent the function

16. anonymous

|dw:1328572233670:dw|

17. TuringTest

we use four rules here really:$\sqrt[b]{x^a}=x^{a/b}$$(xy)^a=x^ay^a$$(x^a)^b=x^{ab}$$x^ax^b=x^{a+b}$$\large\sqrt{x\sqrt[3]{x^2}}=(x\cdot x^{3/2})^{1/2}=x^{1/2}\cdot x^{(3/2)\cdot(1/2)}$

18. TuringTest

or if you prefer$\large\sqrt{x\sqrt[3]{x^2}}=(x\cdot x^{\frac32})^{\frac12}=(x^{1+\frac32})^{\frac12}=x^{(\frac52\cdot\frac12)}=x^{\frac54}$

19. anonymous

gotttttttt it. ty test

20. TuringTest

welcome :D