Given √4096 = 64, evaluate √40.96 + √0.4096 + √0.004096 + √0.00004096.

Introduction / Context:
This problem exploits place-value scaling of a known square. If √4096 = 64, then moving the decimal point in 4096 simply scales the square root by powers of 10. This creates a quick path to compute several square roots without fresh squaring.

Given Data / Assumptions:

• √4096 = 64.
• We need √40.96, √0.4096, √0.004096, and √0.00004096.

Concept / Approach:
If N = 4096, then for any k, √(N / 10^(2k)) = √N / 10^k because dividing inside the root by 10^(2k) divides outside the root by 10^k. Each given decimal equals 4096 divided by a power of 10.

Step-by-Step Solution:
40.96 = 4096 / 100 → √40.96 = 64 / 10 = 6.4.0.4096 = 4096 / 10000 → √0.4096 = 64 / 100 = 0.64.0.004096 = 4096 / 1,000,000 → √0.004096 = 64 / 1000 = 0.064.0.00004096 = 4096 / 100,000,000 → √0.00004096 = 64 / 10000 = 0.0064.Sum = 6.4 + 0.64 + 0.064 + 0.0064 = 7.1104.

Verification / Alternative check:
Note the pattern: each successive term divides the previous square root by 10, ensuring the running total converges and is easy to add accurately.

Why Other Options Are Wrong:
7.09, 7.1014, 7.12, 7.014: Each deviates from the exact decimal scaling and precise addition shown above.

Common Pitfalls:
Miscalculating powers of 10 under the square root; arithmetic slips when summing decimals with mixed place values; rounding prematurely instead of keeping exact tenths, hundredths, thousandths, and ten-thousandths.

Final Answer:
7.1104