The question asks to simplify the following mathematical expression:

$$ \frac{(6.73)^3-(4.38)^3}{(6.73)^2+6.73 \times 4.38+(4.38)^2} $$

This expression is structured around a common algebraic identity, the difference of cubes.

Algebraic Identity for Difference of Cubes

The difference of cubes formula states that:

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

We can use this identity to simplify the given expression.

Applying the Identity to the Problem

Let's identify the values of '$a$' and '$b$' in the given expression:

• $a = 6.73$
• $b = 4.38$

Now, substitute these values into the standard form of the expression:

$$ \frac{a^3 - b^3}{a^2 + ab + b^2} $$

Using the difference of cubes formula, we replace $a^3 - b^3$ in the numerator:

$$ \frac{(a - b)(a^2 + ab + b^2)}{a^2 + ab + b^2} $$

We can see that the term $(a^2 + ab + b^2)$ is present in both the numerator and the denominator. Since $a$ and $b$ are positive numbers, $a^2 + ab + b^2$ will not be zero, allowing us to cancel this term.

After cancellation, the expression simplifies significantly to:

$$ a - b $$

Step-by-Step Calculation

Now, we perform the subtraction using the identified values of $a$ and $b$:

1. Substitute values: $a = 6.73$ and $b = 4.38$.
2. Calculate the difference: $$ 6.73 - 4.38 $$ Performing the subtraction:

   $6.73$
   $- 4.38$
   -------
   $2.35$

   So, $6.73 - 4.38 = 2.35$.

Final Result

The simplified value of the expression $\frac{(6.73)^3-(4.38)^3}{(6.73)^2+6.73 \times 4.38+(4.38)^2}$ is $2.35$. This approach highlights the efficiency of using algebraic identities for simplifying complex calculations.