Given that √18225 = 135, find the value of √182.25 + √1.8225 + √0.018225 + √0.00018225.

Introduction / Context:
This question involves square roots of several decimal numbers that are scaled versions of 18225. We are given that √18225 = 135 and asked to evaluate a sum of related square roots. The problem tests recognition of how square roots behave under scaling and decimal shifts.

Given Data / Assumptions:
- √18225 = 135 is known.
- We must compute S = √182.25 + √1.8225 + √0.018225 + √0.00018225.
- All quantities are positive real numbers.

Concept / Approach:
Each decimal can be written as 18225 multiplied by a power of 10. Since √(k × 18225) = √k × √18225, and √18225 is given, we can express each term in the sum S in terms of 135 and appropriate powers of 10. Then we add the results to find S.

Step-by-Step Solution:
Step 1: Note that 182.25 = 18225 / 100 = 18225 × 10⁻².Step 2: So √182.25 = √(18225 × 10⁻²) = √18225 × √10⁻² = 135 × 10⁻¹ = 13.5.Step 3: Next, 1.8225 = 18225 / 10000 = 18225 × 10⁻⁴, so √1.8225 = 135 × 10⁻² = 1.35.Step 4: For 0.018225, write 0.018225 = 18225 / 10⁶ = 18225 × 10⁻⁶, hence √0.018225 = 135 × 10⁻³ = 0.135.Step 5: For 0.00018225, we have 0.00018225 = 18225 / 10⁸ = 18225 × 10⁻⁸, so √0.00018225 = 135 × 10⁻⁴ = 0.0135.Step 6: Now compute the sum S = 13.5 + 1.35 + 0.135 + 0.0135.Step 7: Add them stepwise: 13.5 + 1.35 = 14.85; 14.85 + 0.135 = 14.985; 14.985 + 0.0135 = 14.9985.

Verification / Alternative check:
We can check each square root using a calculator. For example, √182.25 ≈ 13.5, √1.8225 ≈ 1.35, √0.018225 ≈ 0.135 and √0.00018225 ≈ 0.0135. Adding these approximate values gives 14.9985, matching the exact calculation.

Why Other Options Are Wrong:
The options 1.49985, 149.985 and 1499.85 correspond to shifting the decimal point incorrectly by one or more places. These arise if the power of 10 under the square root is mishandled, for example treating √10⁻² as 10⁻² instead of 10⁻¹.

Common Pitfalls:
Errors often occur when converting decimals to powers of 10 or when taking square roots of these powers. Remember that √10ⁿ = 10^(n/2). For example, √10⁻⁴ = 10⁻², not 10⁻⁴. Misplacing the decimal in intermediate steps can easily lead to answers that differ by factors of 10.

Final Answer:
The value of √182.25 + √1.8225 + √0.018225 + √0.00018225 is 14.9985.