convert to exponential form and simplify.

Question

Assume all variables represent positive nunbers.

 

 

a) (2√x)³/³√x

 

 

b) ⁶√y³z ⋅√y/³√z

Answer 1

I would go with what Andrew suggested. Not sure how to delete posts so guess I will just leave these couple of sentences. 

Answer 2

First off note that roots are basically fractional exponents.  This means that ⁿ√x = x^1/n 

Example   ³√x = x^1/3

 

a.

(2√x)³/³√x = [2³(x^1/2)³]/x^1/3 = 8x^3/2/x^1/3

 

Remember that (x^a)^b = x^a(b) so (x^1/2)³ = x^(1/2)(3) = x^3/2

 

Now note that  xⁿ= 1/x^-n  Which basically means that if you move an exponential term

from the numerator to the denominator or vice versa .. You change the sign of the exponent

 

This give us:

8x^3/2/x^1/3 = 8x^3/2x^1/3 

 

When you multiply exponential numbers you add the exponents ...  x^ax^b = x^a+b  so...

 

8x^3/2x^1/3 = 8x^(3/2+1/3)

 

Now all we have to do is add 3/2 + 1/3 to find the exponent of the x term

3/2 + 1/3 = 9/6 + 2/6 = 11/6

 

8x^(3/2+1/3) = 8x^11/6

 

b.  ⁶√(y³z) - √y/³√z

 

= (⁶√(y³)(⁶√z) - √y/³√z

= (y³)^1/6(z^1/6) - y^1/2/z^1/3

= y^3/6z^1/6 - y^1/2z^-1/3

= y^1/2z^1/6 - y^1/2z^-1/3

 

I can factor out y^1/2 so I get ...

 

= y^1/2(z^1/6-z^-1/3)