For any a, x, y...

Question

For any a, x, y ∈ R define x⊕y = (x³+y³)^1/3 and a⊗x = ³√ax.  Show that a⊗(x⊕y) = (a⊗x)⊕(a⊗y).

Answer 1

Hi Nico,

 

Work on each side of the identity and show that the results are the same.

 

a⊗(x⊕y) 

a⊗ (x³+y³)^1/3

a^1/3(x³+y³)^1/3 (circle x operator indicates cube root of left operand times right operand)

 

 

(a⊗x)⊕(a⊗y)

a^1/3x ⊕ a^1/3y (to perform the circle/plus operation on these two operands sum the cube of both operands and raise the sum to the 1/3 power.

 

So, (ax³ + ay³)^1/3

Factor out the "a" from the expression in ()--

 

[a(x³+y³)]^1/3

 

Applying power of a product properties:

 

a^1/3(x³+y³)^1/3

 

Q.E.D.