What is the value of 105^3 (the cube of 105)?

Introduction / Context:
This problem asks for the cube of 105. Direct multiplication is possible, but using algebraic identities can simplify the calculation. Recognising patterns like (100 + 5)^3 helps in quick mental math and is very useful for competitive exams.

Given Data / Assumptions:

• We need to compute 105^3.
• Options: 1157625, 1175625, 1185625, 1158625.
• We assume standard rules of arithmetic.

Concept / Approach:
We can express 105 as 100 + 5 and use the binomial expansion for (a + b)^3. The identity is (a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3. This allows us to use simpler cubes and squares like 100^3 and 5^3.

Step-by-Step Solution:
Step 1: Write 105^3 as (100 + 5)^3.Step 2: Apply the identity (a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 with a = 100 and b = 5.Step 3: Compute a^3 = 100^3 = 1000000.Step 4: Compute 3a^2b = 3 * 100^2 * 5 = 3 * 10000 * 5 = 150000.Step 5: Compute 3ab^2 = 3 * 100 * 5^2 = 3 * 100 * 25 = 7500.Step 6: Compute b^3 = 5^3 = 125.Step 7: Add all contributions: 1000000 + 150000 = 1150000.Step 8: Add 7500 to get 1157500.Step 9: Finally add 125 to get 1157625.Step 10: Therefore, 105^3 = 1157625.

Verification / Alternative check:
We can quickly check the last digit. Since 5^3 ends in 125, any cube of a number ending in 5 will end in 125. All options end with 625, consistent with this rule, so we rely on the detailed binomial computation above to distinguish the correct value.

Why Other Options Are Wrong:

• 1175625, 1185625, 1158625: These differ by thousands from the correct value 1157625 and result from incorrect middle term calculations or arithmetic errors in adding the contributions from the binomial expansion.

Common Pitfalls:
Errors often occur in computing 3a^2b and 3ab^2, or in adding the large numbers at the end. Writing each step carefully and checking intermediate sums is essential to avoid misplacing digits.

Final Answer:
The value of 105^3 is 1157625.