Value Expression Calculation

The question requires calculating the value of the following mathematical expression:

$$ \frac{(7.03)^{3}-0.027}{(7.03)^{2}+2.109+(0.3)^{2}} $$

We can solve this problem by recognizing a common algebraic identity.

Recognizing Algebraic Identity

Let's examine the structure of the expression. We can identify parts that relate to the difference of cubes formula, which is $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$.

Let $a = 7.03$.

Now, consider the term $0.027$ in the numerator. We need to find a value $b$ such that $b^3 = 0.027$.

We can test $b = 0.3$: $$ (0.3)^3 = 0.3 \times 0.3 \times 0.3 = 0.09 \times 0.3 = 0.027 $$ So, we can set $b = 0.3$.

With $a = 7.03$ and $b = 0.3$, the numerator $(7.03)^{3}-0.027$ can be written as $a^3 - b^3$.

Analyzing the Denominator

Now let's look at the denominator: $(7.03)^{2}+2.109+(0.3)^{2}$.

Using our identified values $a = 7.03$ and $b = 0.3$, we can check if the denominator matches the form $a^2 + ab + b^2$.

• The first term is $(7.03)^2$, which is $a^2$.
• The third term is $(0.3)^2$, which is $b^2$.
• The middle term is $2.109$. Let's calculate $a \times b$: $$ a \times b = 7.03 \times 0.3 = 2.109 $$ This matches the middle term in the denominator.

Therefore, the denominator is exactly $a^2 + ab + b^2$.

Simplifying the Expression

The original expression can be rewritten using $a$ and $b$ as:

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

Now, we substitute the factored form of the numerator, $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$: $$ \frac{(a - b)(a^2 + ab + b^2)}{a^2 + ab + b^2} $$

Since $a = 7.03$ and $b = 0.3$ are positive numbers, the term $a^2 + ab + b^2$ will not be zero. Thus, we can cancel the term $(a^2 + ab + b^2)$ from both the numerator and the denominator.

The expression simplifies to:

$$ a - b $$

Final Calculation

Substitute the values of $a$ and $b$ back into the simplified expression $a - b$:

$$ 7.03 - 0.3 $$

Performing the subtraction:

$$ 7.03 - 0.30 = 6.73 $$

The value of the given expression is $6.73$.