Given `(x^(3)+12x)/(6x^(2)+8)=(y^(3)+27y)/(9y^(2)+27)`. Using componendo and devidendo find x : y.

Answer

To solve the equation

\[ \frac{x^{3}+12x}{6x^{2}+8}=\frac{y^{3}+27y}{9y^{2}+27} \]

using the method of componendo and dividendo, we will follow these steps:

### Step 1: Apply Componendo and Dividendo
We start with the given equation:

\[ \frac{a}{b} = \frac{c}{d} \]

where \( a = x^{3} + 12x \), \( b = 6x^{2} + 8 \), \( c = y^{3} + 27y \), and \( d = 9y^{2} + 27 \).

Using the property of componendo and dividendo, we have:

\[ \frac{a+b}{a-b} = \frac{c+d}{c-d} \]

### Step 2: Calculate \( a + b \) and \( a - b \)
Calculate \( a + b \):

\[ a + b = (x^{3} + 12x) + (6x^{2} + 8) = x^{3} + 6x^{2} + 12x + 8 \]

Calculate \( a - b \):

\[ a - b = (x^{3} + 12x) - (6x^{2} + 8) = x^{3} - 6x^{2} + 12x - 8 \]

### Step 3: Calculate \( c + d \) and \( c - d \)
Calculate \( c + d \):

\[ c + d = (y^{3} + 27y) + (9y^{2} + 27) = y^{3} + 9y^{2} + 27y + 27 \]

Calculate \( c - d \):

\[ c - d = (y^{3} + 27y) - (9y^{2} + 27) = y^{3} - 9y^{2} + 27y - 27 \]

### Step 4: Set Up the New Equation
Now we can set up the new equation:

\[ \frac{x^{3} + 6x^{2} + 12x + 8}{x^{3} - 6x^{2} + 12x - 8} = \frac{y^{3} + 9y^{2} + 27y + 27}{y^{3} - 9y^{2} + 27y - 27} \]

### Step 5: Factor the Expressions
Notice that:

- \( x^{3} + 6x^{2} + 12x + 8 = (x + 2)^{3} \)
- \( x^{3} - 6x^{2} + 12x - 8 = (x - 2)^{3} \)
- \( y^{3} + 9y^{2} + 27y + 27 = (y + 3)^{3} \)
- \( y^{3} - 9y^{2} + 27y - 27 = (y - 3)^{3} \)

Thus, we can rewrite the equation as:

\[ \frac{(x + 2)^{3}}{(x - 2)^{3}} = \frac{(y + 3)^{3}}{(y - 3)^{3}} \]

### Step 6: Take the Cube Root
Taking the cube root of both sides gives us:

\[ \frac{x + 2}{x - 2} = \frac{y + 3}{y - 3} \]

### Step 7: Apply Componendo and Dividendo Again
Now we apply componendo and dividendo again:

\[ \frac{(x + 2) + (x - 2)}{(x + 2) - (x - 2)} = \frac{(y + 3) + (y - 3)}{(y + 3) - (y - 3)} \]

Calculating the left side:

\[ \frac{2x}{4} = \frac{x}{2} \]

Calculating the right side:

\[ \frac{2y}{6} = \frac{y}{3} \]

### Step 8: Set the Two Sides Equal
Setting the two sides equal gives us:

\[ \frac{x}{2} = \frac{y}{3} \]

### Step 9: Solve for \( \frac{x}{y} \)
Cross-multiplying gives:

\[ 3x = 2y \implies \frac{x}{y} = \frac{2}{3} \]

### Final Answer
Thus, the ratio \( x : y \) is

\[ \boxed{\frac{2}{3}} \]