Evaluate the expression (10.3 × 10.3 × 10.3 + 1) ÷ (10.3 × 10.3 − 10.3 + 1) and give the result as a decimal.

Introduction / Context:
This problem uses a special algebraic identity involving cubes and squares. Even though the numbers are decimals, the structure still follows the identity. Recognizing such patterns is very helpful in simplifying complicated looking expressions without heavy computation.

Given Data / Assumptions:

• Numerator: 10.3 × 10.3 × 10.3 + 1, which is 10.3^3 + 1.
• Denominator: 10.3 × 10.3 − 10.3 + 1, which is 10.3^2 − 10.3 + 1.
• We must compute (10.3^3 + 1) / (10.3^2 − 10.3 + 1).

Concept / Approach:
The algebraic identity a^3 + 1 = (a + 1) * (a^2 − a + 1) holds for any real number a, including decimals. Here, a is 10.3. This means the numerator factors nicely, and the denominator is exactly the quadratic factor in that identity. Therefore, dividing the numerator by the denominator simply leaves the factor a + 1, which is 10.3 + 1.

Step-by-Step Solution:
Step 1: Let a = 10.3. Step 2: Recognize the identity a^3 + 1 = (a + 1) * (a^2 − a + 1). Step 3: The numerator is a^3 + 1 = 10.3^3 + 1. Step 4: The denominator is a^2 − a + 1 = 10.3^2 − 10.3 + 1. Step 5: Therefore, (a^3 + 1) / (a^2 − a + 1) = (a + 1) * (a^2 − a + 1) / (a^2 − a + 1). Step 6: The factor (a^2 − a + 1) cancels, leaving simply a + 1. Step 7: Compute a + 1 = 10.3 + 1 = 11.3.

Verification / Alternative check:
We could also perform an approximate numerical check. Compute 10.3^2 ≈ 106.09 and 10.3^3 ≈ 1092.727. Then the numerator is approximately 1092.727 + 1 = 1093.727 and the denominator is about 106.09 − 10.3 + 1 = 96.79. Dividing 1093.727 by 96.79 gives a value near 11.3, which confirms the algebraic result and shows that the identity works correctly for decimals.

Why Other Options Are Wrong:
12.3 would correspond to a different identity or arithmetic result and is larger than the actual value.
13.3 and 14.3 are progressively larger and do not match the simplified expression a + 1 for a = 10.3.

Common Pitfalls:
Learners who do not recognize the identity may try to compute 10.3^3 and 10.3^2 exactly, increasing the risk of decimal errors. Others may misremember the identity a^3 − 1 instead of a^3 + 1. Always verify the pattern of signs in the identity before applying it and remember that it works equally well for decimal values.

Final Answer:
The expression evaluates to 11.3.