Fouth root of 86.21 × 27.34 all divided by 52.18 × 0.0726 using logarithms

Answer 1

To solve \(\sqrt[4]{\frac{86.21 \times 27.34}{52.18 \times 0.0726}}\) using logarithms, we can follow these steps:

1. Calculate the Logarithms: First, we will calculate the logarithm of the numerator and the denominator, and then find their difference.

2. Apply the Fourth Root: After calculating the logarithm of the quotient, we will take the fourth root.

Step 1: Calculate the Logarithms

Let: \[ A = 86.21 \times 27.34 \] \[ B = 52.18 \times 0.0726 \] We want to find: \[ \sqrt[4]{\frac{A}{B}} = \frac{A^{1/4}}{B^{1/4}} = \left( \frac{A}{B} \right)^{1/4} \]

Using logarithms: \[ \log(A) = \log(86.21) + \log(27.34) \] \[ \log(B) = \log(52.18) + \log(0.0726) \]

Now, let's compute each of these logarithms using a calculator:

1. \(\log(86.21) \approx 1.9357\)
2. \(\log(27.34) \approx 1.4368\)
3. \(\log(52.18) \approx 1.7170\)
4. \(\log(0.0726) \approx -1.1384\)

Now calculate: \[ \log(A) \approx 1.9357 + 1.4368 = 3.3725 \] \[ \log(B) \approx 1.7170 - 1.1384 = 0.5786 \]

Step 2: Find the Logarithm of the Quotient

\[ \log\left(\frac{A}{B}\right) = \log(A) - \log(B) \approx 3.3725 - 0.5786 = 2.7939 \]

Step 3: Compute the Fourth Root

To find \(\sqrt[4]{\frac{A}{B}}\), we use: \[ \log\left(\left(\frac{A}{B}\right)^{1/4}\right) = \frac{1}{4} \log\left(\frac{A}{B}\right) \approx \frac{1}{4} \cdot 2.7939 \approx 0.698475 \]

Finally, to find the value, we’ll take the antilogarithm: \[ \frac{A}{B} \approx 10^{0.698475} \approx 5.024 \] Now, taking the fourth root: \[ \sqrt[4]{5.024} \approx \sqrt{\sqrt{5.024}} \approx 1.495 \]

Conclusion

The value of \(\sqrt[4]{\frac{86.21 \times 27.34}{52.18 \times 0.0726}}\) is approximately \( 1.495\).