Question

State how many significant figures are proper in the results of the following calculations: (a)$\left(106.7\right)\left(98.2\right)\mathbf{/}\left(46.210\right)\left(1.01\right)$(b)${\left(18.7\right)}^{\mathbf{2}}$(c)$\left(1.60\times {10}^{-19}\right)\left(3712\right)$.

Short answer

1. The smallest number of significant figures is 3.
2. The smallest number of significant figures is 3.
3. The smallest number of the significant figures is 3.

Step-by-step solution

Defining concept of uncertainty

Uncertainty is the doubt that exists about the result of a measurement, which means how much the values deviate from each other. The precision of the measurement is assessed with significant figures. According to the rule, a significant figure in results must be equal to the quantity with the least number of significant figures.

Given Data

Consider the given data as below.

Calculation (a):$(106.7)(98.2)/(46.210)(1.01)$

Calculation (b):${(18.7)}^2$

​Calculation (c):$(1.60\times {10}^{-19})\left(3712\right)$

Defining the significant figures as proper in the results of (a)

The result 106.7 has 4 significant figures.

The result 98.2 has 3 significant figures.

The result 46.210 has 5 significant figures.

The result 1.01 has 3 significant figures.

The smallest number of significant figures is 3.

The result of 3 significant figures is,-

$(106.7)(98.2)/(46.210)(1.01)=225$

Defining the significant figures as proper in the results of (b)

The given result is ${(18.7)}^2$.

Hence, the result ${(18.7)}^2$has 3 significant figure.

$$\begin{gathered} {(18.7)}^2=18.7\times 18.7 \\ =350 \end{gathered}$$

Hence, the smallest number of significant figures is 3.

Defining the significant figures as proper in the results of (c)

The given result is $\left(1.60\times {10}^{-19}\right)\left(3712\right)$.

Remember when a number is written in scientific notation, only significant figures are the numerical part only. Therefore,-

The number$\left(1.60\times {10}^{-19}\right)$ has 3 significant figures.

And the number $\left(3712\right)$has 4 significant figures.

Here, in the result the smallest number of significant figures is 3. Therefore,-

$\left(1.60\times {10}^{-19}\right)\left(3712\right)=5.94\times {10}^{-16}$