Given √4096 = 64, evaluate the sum: √4096 + √40.96 + √0.004096 = ?

Introduction / Context:This problem checks decimal square-root fluency. Recognizing that decimal shifts in the radicand correspond to dividing or multiplying the square root by 10 helps compute quickly without a calculator.

Given Data / Assumptions:√4096 = 64 is provided.

Concept / Approach:Note that 40.96 = 4096 / 100, so √40.96 = √4096 / 10 = 6.4. Also, 0.004096 = 4096 / 1,000,000, so √0.004096 = √4096 / 1000 = 0.064.

Step-by-Step Solution:

Verification / Alternative check:Square 6.4 to confirm 40.96 and 0.064 to confirm 0.004096: 6.4^2 = 40.96; 0.064^2 = 0.004096.

Why Other Options Are Wrong:70.4 and 71.4 ignore the last term or round incorrectly; 71.104 is an over-add; 69.464 undercounts.

Common Pitfalls:Mistaking decimal place movements when square-rooting decimal numbers; remember √(a/100) = √a / 10 and √(a/10^6) = √a / 1000.

Final Answer:70.464