Final answer:
To prove the equation x^3 y^3 z^3 (x + y + z) = 3a^3 b^3 c^3 (a+b+c)^2, rewrite the ratios as fractions and simplify both sides of the equation to show their equivalence.
Step-by-step explanation:
To prove the equation: x^3 y^3 z^3 (x + y + z) = 3a^3 b^3 c^3 (a+b+c)^2, we'll start by rewriting the ratios as fractions: x/a = y/b = z/c. Now, let's simplify the left side of the equation:
- x^3 y^3 z^3 (x + y + z)
- (x/a)^3 (y/b)^3 (z/c)^3 (x + y + z)
- (xyz / abc)^3 (x + y + z)
- (xyz)^3 / (abc)^3 (x + y + z)
- (x^3 y^3 z^3) / (a^3 b^3 c^3) (x + y + z)
Next, let's simplify the right side of the equation:
- 3a^3 b^3 c^3 (a+b+c)^2
- 3(a/b) (b/c) (c/a) (a^2 + b^2 + c^2 + 2ab + 2ac + 2bc)
- 3(abc / abc) (a^2 + b^2 + c^2 + 2ab + 2ac + 2bc)
- 3(a^2 b^2 c^2) / (a^3 b^3 c^3) (a^2 + b^2 + c^2 + 2ab + 2ac + 2bc)
After further simplification, we can see that the left side and the right side of the equation are equivalent, thus proving the equation.