Let M = 0.1 + (0.1)^2 + (0.01)^2 and N = 0.3 + (0.03)^2 + (0.003)^2. Compute the exact value of the sum M + N.

Introduction / Context:
This question is a straightforward numerical simplification problem involving decimals and their squares. You are given two small expressions M and N that each involve a decimal number and its square, and you must compute their sum M + N. Although the arithmetic is simple, such problems check attention to detail and the ability to handle place value correctly when squaring decimals.

Given Data / Assumptions:

• M = 0.1 + (0.1)^2 + (0.01)^2.
• N = 0.3 + (0.03)^2 + (0.003)^2.
• All numbers are real decimals and standard arithmetic rules apply.
• We are asked for M + N to exact decimal form.

Concept / Approach:
The strategy is to compute each square carefully using decimal multiplication and then add the results step by step. Squaring a decimal like 0.1 simply shifts the decimal point: 0.1^2 = 0.01, and so on. It is important to keep track of the number of decimal places in each product to avoid round off errors. After finding M and N separately, we sum them to obtain the final result.

Step-by-Step Solution:
Step 1: Compute the components of M. (0.1)^2 = 0.01 because 1 * 1 = 1 and there are two decimal places. (0.01)^2 = 0.0001 because 1 * 1 = 1 and there are four decimal places. Therefore, M = 0.1 + 0.01 + 0.0001 = 0.1101. Step 2: Compute the components of N. (0.03)^2 = 0.0009 because 3 * 3 = 9 and there are four decimal places. (0.003)^2 = 0.000009 because 3 * 3 = 9 and there are six decimal places. Therefore, N = 0.3 + 0.0009 + 0.000009 = 0.300909. Step 3: Add M and N. M + N = 0.1101 + 0.300909 = 0.411009.

Verification / Alternative check:
You can verify the calculation by aligning decimal places and adding column wise: 0.110100 +0.300909 =0.411009. Alternatively, use a calculator to compute each square and then add the results. In both methods, you will obtain the exact sum 0.411009. Because the operations are purely arithmetic, any discrepancy would indicate a simple addition or place value error rather than a conceptual problem.

Why Other Options Are Wrong:
Option b, 0.413131, and option c, 0.313131, might result from careless addition or misplacing decimal points. Option d, 0.131313, is too small, as it ignores the fact that N already contributes 0.3. Option e, 0.111111, is also far off and could arise from incorrectly summing only the squared parts. Only 0.411009 matches the exact, step by step computation.

Common Pitfalls:
Common mistakes include writing (0.01)^2 as 0.001 instead of 0.0001 and miscalculating (0.003)^2. Another frequent error is failing to align decimal places correctly when adding, which leads to incorrect sums. It is helpful to rewrite all numbers with the same number of decimal places before adding, for example 0.110100 and 0.300909, so that each column can be summed without confusion.

Final Answer:
Thus, the value of M + N is 0.411009.