Use logarithm tables to evaluate: (36.72 × 0.46 / 185.42)^(1/2 − 1/3)

Question

ScanSolve AI Tutor · Accepted Answer

I have carefully reviewed the image again, including all the cropped versions. The expression for question 15, "Simplify by rationalising the denominator:", is still not visible. There is a blank space where the fraction should be. It seems the expression itself is missing from the question paper.

Since I cannot solve question 15 without the expression, I will solve question 12 for you, which is fully visible.

12. Use logarithm tables to evaluate:

\left(\frac{36.72 \times 0.46}{185.4^2}\right)^{\frac{1}{2} - \frac{1}{3}}

Simplify the exponent.
$\frac{1}{2} - \frac{1}{3} = \frac{3}{6} - \frac{2}{6} = \frac{1}{6}$
So the expression becomes:
$N = \left(\frac{36.72 \times 0.46}{185.4^2}\right)^{\frac{1}{6}}$
Take the logarithm of both sides.
$\log N = \log \left[ \left(\frac{36.72 \times 0.46}{185.4^2}\right)^{\frac{1}{6}} \right]$
Using logarithm properties, $\log(a^b) = b \log a$ and $\log\left(\frac{a \times b}{c}\right) = \log a + \log b - \log c$ :
$\log N = \frac{1}{6} \left[ \log(36.72) + \log(0.46) - \log(185.4^2) \right]$ $\log N = \frac{1}{6} \left[ \log(36.72) + \log(0.46) - 2 \log(185.4) \right]$
Find the logarithms of the numbers using logarithm tables (or a calculator for precision, as tables are not provided).
- $\log(36.72)$ : Characteristic = 1 (since 36.72 is between 10 and 100). Mantissa for 3672 is approximately 0.5649. So, $\log(36.72) \approx 1.5649$ .
- $\log(0.46)$ : Characteristic = $\bar{1}$ (since 0.46 is between 0.1 and 1). Mantissa for 4600 is approximately 0.6628. So, $\log(0.46) \approx \bar{1}.6628$ .
- $\log(185.4)$ : Characteristic = 2 (since 185.4 is between 100 and 1000). Mantissa for 1854 is approximately 0.2681. So, $\log(185.4) \approx 2.2681$ .
Substitute the logarithm values into the equation for $\log N$ .
$\log N = \frac{1}{6} \left[ 1.5649 + (\bar{1}.6628) - 2(2.2681) \right]$ $\log N = \frac{1}{6} \left[ 1.5649 + (-1 + 0.6628) - 4.5362 \right]$ $\log N = \frac{1}{6} \left[ 1.5649 - 0.3372 - 4.5362 \right]$ $\log N = \frac{1}{6} \left[ 1.2277 - 4.5362 \right]$ $\log N = \frac{1}{6} \left[ -3.3085 \right]$ $\log N = -0.551416...$
Convert the negative logarithm to the standard form with a negative characteristic and positive mantissa.
$\log N = -1 + 0.448583...$ $\log N \approx \bar{1}.4486$
Find the antilogarithm of $\bar{1}.4486$ .
Antilog of 0.4486 is approximately 2.810.
Since the characteristic is $\bar{1}$ , the decimal point is placed such that there is one zero before the first non-zero digit.
$N \approx 0.2810$
Therefore, the value is 0.2810.

Use logarithm tables to evaluate: (36.72 * 0.46 / 185.42)^(1/2 - 1/3)

Answer

Step-by-step solution

Simplify the exponent.

Take the logarithm of both sides.

Find the logarithms of the numbers using logarithm tables (or a calculator for precision, as tables are not provided).

Substitute the logarithm values into the equation for $\log N$ .

Convert the negative logarithm to the standard form with a negative characteristic and positive mantissa.

Find the antilogarithm of $\bar{1}.4486$ .

Try one yourself

Still stuck on this one?

More Mathematics Questions