$$ \left( \frac{10.3 \times 10.3 \times 10.3 + 1}{10.3 \times 10.3 - 10.3 + 1} \right) $$ is equal to:

### Concept & Formula This question tests your ability to recognize the sum of cubes expansion, particularly when one of the variables is the number $1$. The underlying algebraic formula is: $$ a^3 + b^3 = (a + b)(a^2 - ab + b^2) $$ ### Step-by-Step Solution Let $a = 10.3$ and $b = 1$. Remember that $1$ can be written as $1^2$ or $1^3$ without changing its value. The expression can be rewritten to reveal its algebraic structure: * Numerator: $(10.3)^3 + (1)^3 \rightarrow a^3 + b^3$ * Denominator: $(10.3)^2 - (10.3 \times 1) + (1)^2 \rightarrow a^2 - ab + b^2$ Construct the algebraic fraction: $$ \frac{a^3 + b^3}{a^2 - ab + b^2} $$ Substitute the expanded form of the numerator: $$ \frac{(a + b)(a^2 - ab + b^2)}{a^2 - ab + b^2} $$ The common trinomial $(a^2 - ab + b^2)$ completely cancels out from both the top and the bottom. The simplified expression is simply $(a + b)$. Finally, calculate the numerical value: $$ 10.3 + 1 = 11.3 $$ ### Exam Strategy & Shortcut The presence of a number cubed plus $1$ in the numerator, divided by a quadratic expression, is a dead giveaway. The shortcut is to simply add the base number and $1$. The calculation $10.3 + 1 = 11.3$ requires zero scratchpad work. ### Common Pitfall Under exam pressure, students sometimes confuse the sum of cubes with the difference of cubes due to the minus sign in the denominator. This leads them to erroneously subtract $1$, arriving at option (a) $9.3$. Always match your operation to the sign connecting the cubed terms in the numerator. ### Final Answer **Therefore, the correct answer is 11.3.**