$ ext{The value of } \frac{(2.75)^3 - (2.25)^3}{(2.75)^2 + 2.75 imes 2.25 + (2.25)^2} ext{ is :}$

Question

$	ext{The value of } \frac{(2.75)^3 - (2.25)^3}{(2.75)^2 + 2.75 	imes 2.25 + (2.25)^2} 	ext{ is :}$

Accepted Answer

To solve the given problem, we need to evaluate the expression:

$$
\frac{(2.75)^3 - (2.25)^3}{(2.75)^2 + 2.75 \times 2.25 + (2.25)^2}
$$

This expression can be simplified using the identity for the difference of cubes:

$$
a^3 - b^3 = (a - b)(a^2 + ab + b^2)
$$

In this case, $a = 2.75$ and $b = 2.25$. Applying the identity, we have:

$$
(2.75)^3 - (2.25)^3 = (2.75 - 2.25)((2.75)^2 + 2.75 \times 2.25 + (2.25)^2)
$$

The denominator of the original expression is exactly the same as the second factor in the identity:

$$
(2.75)^2 + 2.75 \times 2.25 + (2.25)^2
$$

Thus, the expression simplifies to:

$$
\frac{(2.75 - 2.25)((2.75)^2 + 2.75 \times 2.25 + (2.25)^2)}{(2.75)^2 + 2.75 \times 2.25 + (2.25)^2}
$$

The terms $(2.75)^2 + 2.75 \times 2.25 + (2.25)^2$ in the numerator and denominator cancel each other out, leaving:

$$
2.75 - 2.25 = 0.5
$$

Therefore, the value of the expression is 0.5, which corresponds to Option 2.

**Explanation for why other options are incorrect:**

1. **Option 1 (0.3):** The calculation clearly shows that the simplified result is 0.5, not 0.3. The difference $2.75 - 2.25$ is 0.5, not 0.3.

2. **Option 3 (3):** The result of the expression is derived from the difference $2.75 - 2.25$, which is 0.5, not 3. The identity used does not yield a result of 3.

3. **Option 4 (5):** Similarly, the difference $2.75 - 2.25$ is 0.5, not 5. The calculation and simplification process do not support a result of 5.

Thus, the correct answer is indeed Option 2, which is 0.5.