`((6.45)^(3) xx (0.00034)^(1/3) xx (981.4))/((9.37)^(2) xx (8.93)^(1/4) xx (0.0617))`
find the value using log table

Answer

To solve the expression \(\frac{(6.45)^3 \times (0.00034)^{1/3} \times (981.4)}{(9.37)^2 \times (8.93)^{1/4} \times (0.0617)}\) using logarithms, we will follow these steps:

### Step 1: Apply the logarithmic property
We start by taking the logarithm of both sides. Using the property \(\log(\frac{A}{B}) = \log(A) - \log(B)\), we can express our equation as:

\[ \log(x) = \log((6.45)^3 \times (0.00034)^{1/3} \times (981.4)) - \log((9.37)^2 \times (8.93)^{1/4} \times (0.0617)) \]

### Step 2: Expand using logarithmic properties
Next, we can expand the logarithms using the property \(\log(m \times n) = \log(m) + \log(n)\):

\[ \log(x) = \log(6.45^3) + \log(0.00034^{1/3}) + \log(981.4) - \left( \log(9.37^2) + \log(8.93^{1/4}) + \log(0.0617) \right) \]

### Step 3: Apply the power rule
Using the power rule of logarithms \(\log(m^n) = n \cdot \log(m)\), we can simplify further:

\[ \log(x) = 3 \cdot \log(6.45) + \frac{1}{3} \cdot \log(0.00034) + \log(981.4) - \left( 2 \cdot \log(9.37) + \frac{1}{4} \cdot \log(8.93) + \log(0.0617) \right) \]

### Step 4: Substitute values from the log table
Now, we will substitute the values from the logarithm table:

- \(\log(6.45) \approx 0.8103\)
- \(\log(0.00034) \approx -3.468\)
- \(\log(981.4) \approx 2.9918\)
- \(\log(9.37) \approx 0.9710\)
- \(\log(8.93) \approx 0.9511\)
- \(\log(0.0617) \approx -1.234\)

Substituting these values into the equation gives:

\[ \log(x) = 3 \cdot 0.8103 + \frac{1}{3} \cdot (-3.468) + 2.9918 - \left( 2 \cdot 0.9710 + \frac{1}{4} \cdot 0.9511 + (-1.234) \right) \]

### Step 5: Calculate each term
Calculating each term:

1. \(3 \cdot 0.8103 = 2.4309\)
2. \(\frac{1}{3} \cdot (-3.468) \approx -1.156\)
3. \(2.9918\)
4. \(2 \cdot 0.9710 = 1.942\)
5. \(\frac{1}{4} \cdot 0.9511 \approx 0.2378\)

Now substituting these back into the equation:

\[ \log(x) = 2.4309 - 1.156 + 2.9918 - (1.942 + 0.2378 - 1.234) \]

### Step 6: Simplify
Now simplifying:

\[ \log(x) = 2.4309 - 1.156 + 2.9918 - 1.942 - 0.2378 + 1.234 \]

Calculating this step-by-step:

1. \(2.4309 - 1.156 = 1.2749\)
2. \(1.2749 + 2.9918 = 4.2667\)
3. \(4.2667 - 1.942 = 2.3247\)
4. \(2.3247 - 0.2378 = 2.0869\)
5. \(2.0869 + 1.234 = 3.3209\)

Thus, we find:

\[ \log(x) \approx 3.3209 \]

### Step 7: Take the antilogarithm
Finally, we take the antilogarithm to find \(x\):

\[ x = 10^{3.3209} \]

Calculating this gives:

\[ x \approx 2106.0 \]

### Final Answer
The value of the expression is approximately \(2106.0\).