Find value of `(1.01)^(1//2)` according to binomial theorem

Options

A. `1.005`
B. `(1.01)/(2)`
C. `(1.01)/(4)`
D. `1.001`

Answer

To find the value of \( (1.01)^{1/2} \) using the binomial theorem, we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression \( (1.01)^{1/2} \). We can rewrite this as:
\[ (1 + 0.01)^{1/2} \]
Here, we identify \( x = 0.01 \).

### Step 2: Apply the binomial theorem
According to the binomial theorem, for any real number \( n \):
\[ (1 + x)^n \approx 1 + nx + \frac{n(n-1)}{2}x^2 + \ldots \]
For our case, \( n = \frac{1}{2} \) and \( x = 0.01 \). Thus, we can write:
\[ (1 + 0.01)^{1/2} \approx 1 + \frac{1}{2}(0.01) + \frac{\frac{1}{2}(\frac{1}{2} - 1)}{2}(0.01)^2 + \ldots \]

### Step 3: Calculate the first term
The first term is simply:
\[ 1 \]

### Step 4: Calculate the second term
Now, we calculate the second term:
\[ \frac{1}{2}(0.01) = 0.005 \]

### Step 5: Calculate the higher-order term
Next, we calculate the higher-order term. The second term in the binomial expansion is:
\[ \frac{\frac{1}{2} \left( \frac{1}{2} - 1 \right)}{2}(0.01)^2 \]
Calculating this gives:
\[ \frac{\frac{1}{2} \left( -\frac{1}{2} \right)}{2}(0.01)^2 = \frac{-\frac{1}{4}}{2}(0.0001) = -\frac{1}{8}(0.0001) = -0.0000125 \]
Since this term is very small, we can neglect it for our approximation.

### Step 6: Combine the results
Now we can combine the results:
\[ (1.01)^{1/2} \approx 1 + 0.005 - 0.0000125 \approx 1.005 \]

### Final Answer
Thus, the value of \( (1.01)^{1/2} \) according to the binomial theorem is approximately:
\[ \boxed{1.005} \]