Find an approximate value of the following corrected to 4 decimal places.
`root(7)(127)`

Answer

To find the approximate value of \( \sqrt[7]{127} \) corrected to four decimal places, we can follow these steps:

### Step 1: Rewrite the expression
We start by rewriting \( \sqrt[7]{127} \) in exponential form:
\[ \sqrt[7]{127} = 127^{\frac{1}{7}}. \]
Next, we can express 127 as \( 128 - 1 \):
\[ 127^{\frac{1}{7}} = (128 - 1)^{\frac{1}{7}}. \]

### Step 2: Use the Binomial Expansion
Using the Binomial theorem, we can expand \( (a - b)^n \) where \( a = 128 \), \( b = 1 \), and \( n = \frac{1}{7} \):
\[ (128 - 1)^{\frac{1}{7}} = 128^{\frac{1}{7}} \left(1 - \frac{1}{128}\right)^{\frac{1}{7}}. \]

### Step 3: Calculate \( 128^{\frac{1}{7}} \)
We know that \( 128 = 2^7 \), therefore:
\[ 128^{\frac{1}{7}} = 2. \]

### Step 4: Expand \( \left(1 - \frac{1}{128}\right)^{\frac{1}{7}} \)
Now we need to expand \( \left(1 - \frac{1}{128}\right)^{\frac{1}{7}} \) using the Binomial expansion:
\[ \left(1 - x\right)^p \approx 1 - px + \frac{p(p-1)}{2!}x^2 \quad \text{for small } x. \]
Here, \( p = \frac{1}{7} \) and \( x = \frac{1}{128} \):
\[ \left(1 - \frac{1}{128}\right)^{\frac{1}{7}} \approx 1 - \frac{1}{7} \cdot \frac{1}{128} + \frac{\frac{1}{7} \left(\frac{1}{7} - 1\right)}{2} \left(\frac{1}{128}\right)^2. \]

### Step 5: Calculate the terms
Calculating the first term:
\[ 1 - \frac{1}{7 \cdot 128} = 1 - \frac{1}{896} \approx 1 - 0.00111607 \approx 0.99888393. \]
Calculating the second term:
\[ \frac{\frac{1}{7} \cdot \left(\frac{1}{7} - 1\right)}{2} \left(\frac{1}{128}\right)^2 = \frac{\frac{1}{7} \cdot -\frac{6}{7}}{2} \cdot \frac{1}{16384} = -\frac{6}{14 \cdot 16384} \approx -0.000026. \]
So, we can ignore the second term for our approximation since it is very small.

### Step 6: Combine the results
Now we can combine our results:
\[ \left(1 - \frac{1}{128}\right)^{\frac{1}{7}} \approx 0.99888393. \]
Thus, we have:
\[ \sqrt[7]{127} \approx 2 \cdot 0.99888393 \approx 1.99776786. \]

### Step 7: Round to four decimal places
Finally, rounding \( 1.99776786 \) to four decimal places gives:
\[ \sqrt[7]{127} \approx 1.9978. \]

### Final Answer
The approximate value of \( \sqrt[7]{127} \) corrected to four decimal places is:
\[ \boxed{1.9978}. \]