Using logarithm tables to evaluate: 846.2^(2/3) * sqrt(5436) / cbrt(462.4)

Using the properties $\log(AB) = \log A + \log B$, $\log(A/B) = \log A - \log B$, and $\log(A^n) = n \log A$:

• For 846.2: The characteristic (integer part) is $3-1=2$. The mantissa (decimal part) for 8462 from logarithm tables is approximately 0.9275. So, $\log_{10}(846.2) = 2.9275$.

• For 5436: The characteristic is $4-1=3$. The mantissa for 5436 from logarithm tables is approximately 0.7353. So, $\log_{10}(5436) = 3.7353$.

• For 462.4: The characteristic is $3-1=2$. The mantissa for 4624 from logarithm tables is approximately 0.6650. So, $\log_{10}(462.4) = 2.6650$.

First, add the positive terms:

Now, subtract the negative term:

Rounding the mantissa to four decimal places for antilog table lookup:

The characteristic is 3, which means the number $N$ will have $3+1=4$ digits before the decimal point.
The mantissa is 0.9068.
Looking up 0.9068 in antilogarithm tables, we find the value is approximately 8.069.
Therefore, $N \approx 8069$.

The evaluated expression is 8069.