The sum of the coefficients in the binomial expansion of `(1/x +2x)^6` is equal to :

Options

A. 1024
B. 729
C. 243
D. 512

Answer

To find the sum of the coefficients in the binomial expansion of \((\frac{1}{x} + 2x)^6\), we can follow these steps:

### Step 1: Identify the expression
The expression we need to expand is:
\[ \left(\frac{1}{x} + 2x\right)^6 \]

### Step 2: Use the Binomial Theorem
According to the Binomial Theorem, the expansion of \((a + b)^n\) is given by:
\[ \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \]
In our case, \(a = \frac{1}{x}\), \(b = 2x\), and \(n = 6\).

### Step 3: Substitute values into the binomial expansion
Substituting the values into the binomial expansion, we get:
\[ \sum_{k=0}^{6} \binom{6}{k} \left(\frac{1}{x}\right)^{6-k} (2x)^k \]

### Step 4: Simplify the terms
This simplifies to:
\[ \sum_{k=0}^{6} \binom{6}{k} \left(\frac{1}{x}\right)^{6-k} (2^k x^k) \]
\[ = \sum_{k=0}^{6} \binom{6}{k} 2^k \frac{x^k}{x^{6-k}} \]
\[ = \sum_{k=0}^{6} \binom{6}{k} 2^k x^{2k - 6} \]

### Step 5: Find the sum of the coefficients
To find the sum of the coefficients, we can substitute \(x = 1\):
\[ \sum_{k=0}^{6} \binom{6}{k} 2^k (1)^{2k - 6} = \sum_{k=0}^{6} \binom{6}{k} 2^k \]

### Step 6: Calculate the sum using the binomial theorem
The sum \(\sum_{k=0}^{n} \binom{n}{k} x^k\) is equal to \((1 + x)^n\). Therefore, we can write:
\[ \sum_{k=0}^{6} \binom{6}{k} 2^k = (1 + 2)^6 = 3^6 \]

### Step 7: Calculate \(3^6\)
Now we calculate \(3^6\):
\[ 3^6 = 729 \]

### Final Answer
Thus, the sum of the coefficients in the binomial expansion of \((\frac{1}{x} + 2x)^6\) is:
\[ \boxed{729} \]