If `M = 0.1 + (0.1)^(2) + (0.01)^(2) and N = 0.3 + (0.03)^(2) + (0.003)^(2)`, then what is the value of M + N ?

Options

A. 0.411009
B. `0.413131`
C. `0.313131`
D. `0.131313`

Answer

To find the value of \( M + N \), we first need to calculate \( M \) and \( N \) separately.

### Step 1: Calculate \( M \)

Given:
\[ M = 0.1 + (0.1)^2 + (0.01)^2 \]

Calculating each term:
1. \( 0.1 = 0.1 \)
2. \( (0.1)^2 = 0.01 \)
3. \( (0.01)^2 = 0.0001 \)

Now, sum these values:
\[ M = 0.1 + 0.01 + 0.0001 \]

Aligning the decimal points for addition:
```
0.1000
+ 0.0100
+ 0.0001
---------
0.1101
```
Thus,
\[ M = 0.1101 \]

### Step 2: Calculate \( N \)

Given:
\[ N = 0.3 + (0.03)^2 + (0.003)^2 \]

Calculating each term:
1. \( 0.3 = 0.3 \)
2. \( (0.03)^2 = 0.0009 \)
3. \( (0.003)^2 = 0.000009 \)

Now, sum these values:
\[ N = 0.3 + 0.0009 + 0.000009 \]

Aligning the decimal points for addition:
```
0.300000
+ 0.000900
+ 0.000009
-----------
0.300909
```
Thus,
\[ N = 0.300909 \]

### Step 3: Calculate \( M + N \)

Now we need to find \( M + N \):
\[ M + N = 0.1101 + 0.300909 \]

Aligning the decimal points for addition:
```
0.110100
+ 0.300909
-----------
0.411009
```
Thus,
\[ M + N = 0.411009 \]

### Final Answer
The value of \( M + N \) is \( 0.411009 \).

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