`((4.53 - 3.07)^(2))/((3.07 - 2.15) (2.15 - 4.53)) + ((3.07 - 2.15)^(2))/((2.15 - 4.53) (4.53 - 3.07)) + ((2.15 - 4.53)^(2))/((4.53 - 3.07) (3.07 - 2.15))` is simplified to

Options

A. 0
B. 1
C. 2
D. 3

Answer

To simplify the expression

\[ \frac{(4.53 - 3.07)^{2}}{(3.07 - 2.15)(2.15 - 4.53)} + \frac{(3.07 - 2.15)^{2}}{(2.15 - 4.53)(4.53 - 3.07)} + \frac{(2.15 - 4.53)^{2}}{(4.53 - 3.07)(3.07 - 2.15)}, \]

we will follow these steps:

### Step 1: Identify the terms
Let:
- \( a = 4.53 - 3.07 \)
- \( b = 3.07 - 2.15 \)
- \( c = 2.15 - 4.53 \)

### Step 2: Rewrite the expression
The expression can be rewritten as:

\[ \frac{a^2}{b \cdot c} + \frac{b^2}{c \cdot a} + \frac{c^2}{a \cdot b} \]

### Step 3: Find the common denominator
The common denominator for the three fractions is \( a \cdot b \cdot c \). Thus, we can express the entire expression as:

\[ \frac{a^2 \cdot a + b^2 \cdot b + c^2 \cdot c}{a \cdot b \cdot c} \]

### Step 4: Simplify the numerator
Now, we can simplify the numerator:

\[ a^3 + b^3 + c^3 \]

### Step 5: Use the identity for cubes
We can use the identity:

\[ a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - ac - bc) \]

Since \( a + b + c = (4.53 - 3.07) + (3.07 - 2.15) + (2.15 - 4.53) = 0 \), we have:

\[ a^3 + b^3 + c^3 = 3abc \]

### Step 6: Calculate \( abc \)
Now, we calculate \( abc \):

\[ abc = (4.53 - 3.07)(3.07 - 2.15)(2.15 - 4.53) \]

Calculating each term:
- \( 4.53 - 3.07 = 1.46 \)
- \( 3.07 - 2.15 = 0.92 \)
- \( 2.15 - 4.53 = -2.38 \)

Thus,

\[ abc = 1.46 \cdot 0.92 \cdot (-2.38) \]

### Step 7: Substitute back into the expression
Now substituting back into our expression:

\[ \frac{3abc}{abc} = 3 \]

### Final Answer:
The simplified value of the expression is:

\[ \boxed{3} \]