The area of a reactangle is `x ^(2) + 12 xy + 27 y ^(2)` and its length is (`x + 9y`). Find the breadth of the reactangle.

Answer

To find the breadth of the rectangle given the area and length, we can follow these steps:

### Step 1: Write down the formula for the area of a rectangle.
The area \( A \) of a rectangle is given by the formula:
\[ A = \text{length} \times \text{breadth} \]

### Step 2: Substitute the known values into the area formula.
We know the area \( A = x^2 + 12xy + 27y^2 \) and the length \( l = x + 9y \). Therefore, we can write:
\[ x^2 + 12xy + 27y^2 = (x + 9y) \times \text{breadth} \]

### Step 3: Rearrange the equation to find the breadth.
To find the breadth \( b \), we can rearrange the equation:
\[ \text{breadth} = \frac{x^2 + 12xy + 27y^2}{x + 9y} \]

### Step 4: Factor the numerator.
Next, we need to factor the expression \( x^2 + 12xy + 27y^2 \). We can look for two numbers that add up to 12 and multiply to \( 27 \). The numbers are 9 and 3. Thus, we can rewrite the expression as:
\[ x^2 + 9xy + 3xy + 27y^2 = x^2 + 9xy + 3xy + 27y^2 \]
This can be grouped as:
\[ = (x^2 + 9xy) + (3xy + 27y^2) \]
Factoring out common terms gives:
\[ = x(x + 9y) + 3y(x + 9y) = (x + 9y)(x + 3y) \]

### Step 5: Substitute the factored form back into the breadth formula.
Now substituting back into the breadth formula:
\[ \text{breadth} = \frac{(x + 9y)(x + 3y)}{(x + 9y)} \]

### Step 6: Simplify the expression.
Since \( (x + 9y) \) in the numerator and denominator cancels out (assuming \( x + 9y \neq 0 \)):
\[ \text{breadth} = x + 3y \]

### Final Answer:
Thus, the breadth of the rectangle is:
\[ \text{breadth} = x + 3y \]

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