If `m`, `n` are positive integers and `m+nsqrt(2)=sqrt(41+24sqrt(2))`, then `(m+n)` is equal to (a) 5 (b) 6 (c) 7 (d) 8

Options

A. `5`
B. `6`
C. `7`
D. `8`

Explanation

To solve the equation \( m + n\sqrt{2} = \sqrt{41 + 24\sqrt{2}} \), where \( m \) and \( n \) are positive integers, we will follow these steps:

### Step 1: Square both sides
We start by squaring both sides of the equation to eliminate the square root on the right side.

\[ (m + n\sqrt{2})^2 = (\sqrt{41 + 24\sqrt{2}})^2 \]

This gives us:

\[ m^2 + 2mn\sqrt{2} + 2n^2 = 41 + 24\sqrt{2} \]

### Step 2: Separate the rational and irrational parts
Now, we can separate the rational and irrational parts of the equation:

1. The rational part: \( m^2 + 2n^2 = 41 \)
2. The irrational part: \( 2mn = 24 \)

### Step 3: Solve the equations
From the irrational part, we can simplify:

\[ mn = 12 \]

Now we have a system of equations:

1. \( m^2 + 2n^2 = 41 \) (Equation 1)
2. \( mn = 12 \) (Equation 2)

### Step 4: Express one variable in terms of the other
From Equation 2, we can express \( m \) in terms of \( n \):

\[ m = \frac{12}{n} \]

### Step 5: Substitute into the first equation
Now, substitute \( m \) into Equation 1:

\[ \left(\frac{12}{n}\right)^2 + 2n^2 = 41 \]

This simplifies to:

\[ \frac{144}{n^2} + 2n^2 = 41 \]

### Step 6: Multiply through by \( n^2 \) to eliminate the fraction
Multiplying through by \( n^2 \):

\[ 144 + 2n^4 = 41n^2 \]

Rearranging gives us:

\[ 2n^4 - 41n^2 + 144 = 0 \]

### Step 7: Let \( x = n^2 \)
Let \( x = n^2 \), then we have a quadratic equation:

\[ 2x^2 - 41x + 144 = 0 \]

### Step 8: Solve the quadratic equation
Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):

\[ x = \frac{41 \pm \sqrt{(-41)^2 - 4 \cdot 2 \cdot 144}}{2 \cdot 2} \]

Calculating the discriminant:

\[ b^2 - 4ac = 1681 - 1152 = 529 \]

So,

\[ x = \frac{41 \pm 23}{4} \]

Calculating the two possible values for \( x \):

1. \( x = \frac{64}{4} = 16 \)
2. \( x = \frac{18}{4} = 4.5 \) (not valid since \( n^2 \) must be an integer)

Thus, \( n^2 = 16 \) implies \( n = 4 \).

### Step 9: Find \( m \)
Now substituting \( n = 4 \) back into \( mn = 12 \):

\[ m \cdot 4 = 12 \implies m = 3 \]

### Step 10: Calculate \( m + n \)
Finally, we find:

\[ m + n = 3 + 4 = 7 \]

Thus, the value of \( m + n \) is:

\[ \boxed{7} \]