Find the value of `(627)^(1//4) ` correct to 4 decimals places

Answer

To find the value of \( (627)^{\frac{1}{4}} \) correct to four decimal places, we can use the binomial theorem and some calculus concepts. Here’s a step-by-step solution:

### Step 1: Rewrite the expression
We can express \( 627 \) as \( 625 + 2 \). Thus, we rewrite the expression:
\[ (627)^{\frac{1}{4}} = (625 + 2)^{\frac{1}{4}} \]

### Step 2: Define a function
Let \( y = f(x) = x^{\frac{1}{4}} \). We will use the derivative of this function to approximate the value.

### Step 3: Calculate the derivative
The derivative of \( f(x) \) is:
\[ f'(x) = \frac{1}{4} x^{-\frac{3}{4}} \]

### Step 4: Choose \( x \) and \( \Delta x \)
We choose \( x = 625 \) (since \( 625 \) is a perfect fourth power) and \( \Delta x = 2 \) (because \( 627 - 625 = 2 \)).

### Step 5: Calculate \( f(625) \)
Now, we calculate:
\[ f(625) = 625^{\frac{1}{4}} = 5 \]

### Step 6: Calculate \( f'(625) \)
Now we find \( f'(625) \):
\[ f'(625) = \frac{1}{4} \cdot 625^{-\frac{3}{4}} = \frac{1}{4} \cdot \frac{1}{(625^{\frac{3}{4}})} \]
Calculating \( 625^{\frac{3}{4}} \):
\[ 625^{\frac{3}{4}} = (5^4)^{\frac{3}{4}} = 5^3 = 125 \]
Thus,
\[ f'(625) = \frac{1}{4} \cdot \frac{1}{125} = \frac{1}{500} \]

### Step 7: Use the linear approximation
Using the linear approximation:
\[ \Delta y \approx f'(625) \cdot \Delta x \]
Substituting the values:
\[ \Delta y \approx \frac{1}{500} \cdot 2 = \frac{2}{500} = 0.004 \]

### Step 8: Calculate \( (627)^{\frac{1}{4}} \)
Now we can find:
\[ (627)^{\frac{1}{4}} \approx f(625) + \Delta y = 5 + 0.004 = 5.004 \]

### Final Answer
Thus, the value of \( (627)^{\frac{1}{4}} \) correct to four decimal places is:
\[ \boxed{5.0040} \]