If m and n are the smallest positive integers satisfying the relation `(2CiS pi/6)^m=(4CiSpi/4)^n` , where `i = sqrt(-1), (m+ n)` equals to

Options

A. (a)60
B. (b)72
C. (c)96
D. (d)36

Answer

To solve the equation \( (2 \text{C} i \text{s} \frac{\pi}{6})^m = (4 \text{C} i \text{s} \frac{\pi}{4})^n \), where \( i = \sqrt{-1} \), we can follow these steps:

### Step 1: Rewrite the expressions using Euler's formula
Using Euler's formula, we have:
\[ \text{C} i \text{s} x = \cos x + i \sin x = e^{ix} \]
Thus, we can rewrite the terms:
\[ 2 \text{C} i \text{s} \frac{\pi}{6} = 2 e^{i \frac{\pi}{6}} \quad \text{and} \quad 4 \text{C} i \text{s} \frac{\pi}{4} = 4 e^{i \frac{\pi}{4}} \]

### Step 2: Substitute the rewritten expressions into the equation
Now substituting these into the original equation:
\[ (2 e^{i \frac{\pi}{6}})^m = (4 e^{i \frac{\pi}{4}})^n \]
This simplifies to:
\[ 2^m e^{i \frac{m \pi}{6}} = 4^n e^{i \frac{n \pi}{4}} \]

### Step 3: Express \( 4^n \) in terms of base 2
Since \( 4 = 2^2 \), we can rewrite \( 4^n \) as:
\[ 4^n = (2^2)^n = 2^{2n} \]
Thus, the equation becomes:
\[ 2^m e^{i \frac{m \pi}{6}} = 2^{2n} e^{i \frac{n \pi}{4}} \]

### Step 4: Equate the magnitudes and the arguments
From the magnitudes, we have:
\[ 2^m = 2^{2n} \implies m = 2n \]

From the arguments, we have:
\[ \frac{m \pi}{6} = \frac{n \pi}{4} \]
Dividing both sides by \( \pi \):
\[ \frac{m}{6} = \frac{n}{4} \]
Cross-multiplying gives:
\[ 4m = 6n \implies 2m = 3n \]

### Step 5: Solve the system of equations
Now we have two equations:
1. \( m = 2n \)
2. \( 2m = 3n \)

Substituting \( m = 2n \) into the second equation:
\[ 2(2n) = 3n \implies 4n = 3n \implies n = 0 \]
This is not valid since we need positive integers. Let's express \( n \) in terms of \( m \):
From \( m = 2n \), we can substitute \( n = \frac{m}{2} \) into \( 2m = 3n \):
\[ 2m = 3\left(\frac{m}{2}\right) \implies 2m = \frac{3m}{2} \implies 4m = 3m \implies m = 0 \]
This is again invalid.

### Step 6: Find the least common multiples
To find the smallest positive integers \( m \) and \( n \) that satisfy both equations, we can set:
Let \( m = 12 \) (the least common multiple of 6 and 4) then:
\[ n = \frac{m}{2} = \frac{12}{2} = 6 \]

### Step 7: Calculate \( m + n \)
Now we find \( m + n \):
\[ m + n = 12 + 6 = 18 \]

### Final Answer
Thus, the value of \( m + n \) is \( 18 \).

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