Exponents and Polynomials

Are these three different problems? What are the instructions?

Eliz.
 
(3xy^2)3 (xy^3)=

when you do the y^2 times y^3 it is y^5 not 2y^6
 
(3xy^2)^3 (xy^3)

this is quite different having the power of 3....

(27 x^3 y^6) (x y^3)

27 x^4 y^9
 
(3xy^2)^3 (xy^3)

apply the power of 3 to all the items within the brackets as per this

(3xy^2)^3

3^3 x^3 (y^2)^3

27 x^3 y^6 (because you multiply powers when powers are raised to powers)

next step .....

27 x^3 y^6 x y^3
27 x^4 y^9
 
going back to this step
27 x^3 y^6 x y^3

gather the terms together...

27 (x^3)(x) (y^6)(y^3)

now any value by itself is actually raised to the power of 1 ie x = x^1

27 (x^3) (x^1) (y^6)(y^3)
27 (x^(3+1) ) ( y ^ (6+3) )

27 x^4 y^9
 
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