If ` 3 sqrt( a) + 3 sqrt( b) = 3 sqrt( c)` , then the simplest value of ` ( a + b - c) ^(3) + 27` abc is

Options

A. `-1`
B. 3
C. `-3`
D. 0

Answer

To solve the equation \( 3 \sqrt{a} + 3 \sqrt{b} = 3 \sqrt{c} \), we can simplify it step by step.

### Step 1: Simplify the equation
First, we can divide the entire equation by 3:
\[ \sqrt{a} + \sqrt{b} = \sqrt{c} \]

### Step 2: Cube both sides
Next, we will cube both sides of the equation:
\[ (\sqrt{a} + \sqrt{b})^3 = (\sqrt{c})^3 \]
This simplifies to:
\[ \sqrt{a}^3 + 3\sqrt{a}^2\sqrt{b} + 3\sqrt{a}\sqrt{b}^2 + \sqrt{b}^3 = c \]
Since \(\sqrt{a}^3 = a\) and \(\sqrt{b}^3 = b\), we can rewrite the equation as:
\[ a + b + 3\sqrt{ab}(\sqrt{a} + \sqrt{b}) = c \]
But since \(\sqrt{a} + \sqrt{b} = \sqrt{c}\):
\[ a + b + 3\sqrt{ab}\sqrt{c} = c \]

### Step 3: Rearranging the equation
Now, we rearrange the equation to isolate \(c\):
\[ a + b - c + 3\sqrt{ab}\sqrt{c} = 0 \]
This implies:
\[ a + b - c = -3\sqrt{ab}\sqrt{c} \]

### Step 4: Finding \( (a + b - c)^3 + 27abc \)
Let \( x = a + b - c \). Then, we have:
\[ x = -3\sqrt{ab}\sqrt{c} \]
Cubing both sides gives:
\[ x^3 = (-3\sqrt{ab}\sqrt{c})^3 = -27abc \]
Thus:
\[ x^3 + 27abc = 0 \]

### Step 5: Conclusion
The simplest value of \( (a + b - c)^3 + 27abc \) is:
\[ 0 \]