Evaluate : `(0.0043)^(1//7)`

Answer

To evaluate \( (0.0043)^{\frac{1}{7}} \), we will follow these steps:

### Step 1: Set up the equation
Let \( x = (0.0043)^{\frac{1}{7}} \).

### Step 2: Take the natural logarithm of both sides
Taking the natural logarithm gives us:
\[ \ln x = \ln((0.0043)^{\frac{1}{7}}) \]

### Step 3: Use the logarithmic property
Using the property of logarithms that states \( \ln(a^b) = b \cdot \ln a \), we can rewrite the equation as:
\[ \ln x = \frac{1}{7} \ln(0.0043) \]

### Step 4: Rewrite \( 0.0043 \)
We can express \( 0.0043 \) in terms of its components:
\[ 0.0043 = \frac{43}{10000} \]

### Step 5: Apply the logarithmic property for fractions
Using the property \( \ln\left(\frac{a}{b}\right) = \ln a - \ln b \), we can write:
\[ \ln(0.0043) = \ln(43) - \ln(10000) \]

### Step 6: Calculate \( \ln(10000) \)
Since \( 10000 = 10^4 \), we have:
\[ \ln(10000) = 4 \ln(10) \]
Given that \( \ln(10) \approx 2.302 \), we find:
\[ \ln(10000) \approx 4 \times 2.302 \approx 9.208 \]

### Step 7: Substitute back into the equation
Now substituting back, we have:
\[ \ln(0.0043) = \ln(43) - 9.208 \]

### Step 8: Approximate \( \ln(43) \)
Using a known value, we can approximate \( \ln(43) \approx 3.761 \). Thus:
\[ \ln(0.0043) \approx 3.761 - 9.208 \approx -5.447 \]

### Step 9: Substitute into the equation for \( \ln x \)
Now substituting this back into our equation for \( \ln x \):
\[ \ln x = \frac{1}{7} \cdot (-5.447) \approx -0.778 \]

### Step 10: Find \( x \) by taking the antilogarithm
To find \( x \), we take the antilogarithm:
\[ x = e^{-0.778} \]

### Step 11: Calculate \( e^{-0.778} \)
Using a calculator or approximation, we find:
\[ e^{-0.778} \approx 0.459 \]

### Final Answer
Thus, the value of \( (0.0043)^{\frac{1}{7}} \approx 0.459 \).

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