Question

If $\dfrac{{{a^3} + 3a{b^2}}}{{3{a^2}b + {b^3}}} = \dfrac{{{x^3} + 3x{y^2}}}{{3{x^2}y + {y^3}}}$, then
A) $bx = ay$
B) $by = ax$
C) ${b^2}y = {a^2}x$
D) ${b^2}x = {a^2}y$

Answer 1

Hint: According to given in the question we have to simplify the given expression $\dfrac{{{a^3} + 3a{b^2}}}{{3{a^2}b + {b^3}}} = \dfrac{{{x^3} + 3x{y^2}}}{{3{x^2}y + {y^3}}}$ so, first of all we have to apply the componendo and dividendo rule in the both sides of the given expression.

Formula used: $
   \Rightarrow {(a + b)^3} = {a^3} + {b^3} + 3{a^2}b + 3a{b^2}..............(A) \\
   \Rightarrow {(a - b)^3} = {a^3} - {b^3} - 3{a^2}b + 3a{b^2}..............(B)
 $

Complete step-by-step answer:
Step 1: First of all we have to apply the componendo and dividendo rule in the both sides of the expression. Hence,
\[ \Rightarrow \dfrac{{{a^3} + 3a{b^2} + 3{a^2}b + {b^3}}}{{{a^3} + 3a{b^2} - {b^3} - 3{a^2}b}} = \dfrac{{{x^3} + 3x{y^2} + 3{x^2}y + {y^3}}}{{{x^3} + 3x{y^2} - {y^3} - 3{x^2}y}}\]…………….(1)
Step 2: Now, to solve the expression (1) as obtained in the solution step 1 we have to use the formulas (A) and (B) as mentioned in the solution hint. Hence,
$ \Rightarrow \dfrac{{{{(a + b)}^3}}}{{{{(a - b)}^3}}} = \dfrac{{{{(x + y)}^3}}}{{{{(x - y)}^3}}}$
Step 3: On eliminating the cube roots from the both sides of the expression as obtained in the solution step 2. Hence,
$ \Rightarrow \dfrac{{(a + b)}}{{(a - b)}} = \dfrac{{(x + y)}}{{(x - y)}}$
Step 4: On applying cross multiplication in the expression as obtained in the solution step 3. Hence,
\[
   = (a + b)(x - y) = (a - b)(x + y) \\
   = ax - ay + bx - by = ax + ay - bx - by
 \]
On eliminating the terms which can be eliminated from the expression as obtained just above, $
   \Rightarrow - ay + bx = ay - bx \\
   \Rightarrow 2ay = 2bx
 $
$ \Rightarrow ay = bx$
Final solution: Hence, with the help of the formula (A) and formula (B) we have simplified the given expression which is $ay = bx$.

Therefore the correct option is (A).

Note: For these types of expression it is necessary to apply componendo and dividendo rules to eliminate the terms which can be eliminated.
To make the expression simple we have to make the terms as they can be converted into a formula or a simpler format.