Chapter 1, Problem 42E

Perform the following mathematical operations, and express the result to the correct number of significant figures.

1. a. 6.022 X 10^-23 × 1.05 x 10²
2. b. $\frac{6.6262\times {10}^{-34}\times 2.998\times {10}^8}{2.54\times {10}^{-9}}$
3. c. 1.285 × 10⁻² + 1.24 × 10⁻³ + 1.879 × 10⁻¹
4. d. $\frac{\left(1.00866\text{ - 1}\text{.00728}\right)}{6.02205\times {\text{ 10}}^{\text{23}}}$
5. e. $\frac{9.875\times {\text{10}}^{\text{2}}-9.795\times {10}^2}{9.875\times {10}^2}\times 100(100\text{ is exact})$
6. f. $\frac{9.42\times {\text{10}}^{\text{2}}+8.234\times {10}^2+1.625\times {10}^3}{3}(3\text{ is exact})$

Interpretation Introduction

Interpretation:

For the given set of mathematical calculations the answer has to be given in appropriate number of significant figures.

Concept introduction:

Significant figures: The digits having a meaning in a given number are called as significant figures.

Significant figures – calculation rules

• Significant figure is a non-zero number (4443.2 has five significant figures)
• Zeros placement in between non-zero number are significant figures (40005 has five significant figures).
• Zeros before the non-zero number are not significant number (0.00040005 has five significant figures).
• Zeros after the non-zero number are significant in case of that number contains decimal point (4.5000 has five significant figures).
• Zeros after the non-zero number may or may not be a significant figure in case of that number does not have a decimal point (500 may have 1, 2 or 3 significant figures).  In this cases use scientific notation to avoid ambiguity.
• The placement of decimal at the trailing zero is indicates the zero is significant (550. has three significant figures).

Answer to Problem 42E

The answer is $\text{6}\text{.32}\text{×}{\text{10}}^{\text{25}}$.

Explanation of Solution

Given data:

$\text{6}\text{.022}\text{×}{\text{10}}^{\text{23}}\text{×}\text{1}\text{.05}\text{×}{\text{10}}^{\text{2}}$

Take the multiplication of the given values

$\begin{array}{l}\text{6}\text{.022}\text{×}{\text{10}}^{\text{23}}\text{×}\text{1}\text{.05}\text{×}{\text{10}}^{\text{2}}\\ \text{=}\text{6}\text{.022}\text{×}\text{1}\text{.05}\text{×}{\text{10}}^{\text{25}}\\ \text{=}\text{6}\text{.3231}\text{×}{\text{10}}^{\text{25}}\end{array}$

The answer is $\text{6}\text{.32}\text{×}{\text{10}}^{\text{25}}$.

The result is $\text{6}\text{.3231}\text{×}{\text{10}}^{\text{25}}$.  Here the limiting term $\text{1}\text{.05}$ has two decimal places.  Hence, the obtained result was rounded off to get the final result with appropriate number of significant figures.

Interpretation Introduction

Interpretation:

For the given set of mathematical calculations the answer has to be given in appropriate number of significant figures.

Concept introduction:

Significant figures: The digits having a meaning in a given number are called as significant figures.

Significant figures – calculation rules

• Significant figure is a non-zero number (4443.2 has five significant figures)
• Zeros placement in between non-zero number are significant figures (40005 has five significant figures).
• Zeros before the non-zero number are not significant number (0.00040005 has five significant figures).
• Zeros after the non-zero number are significant in case of that number contains decimal point (4.5000 has five significant figures).
• Zeros after the non-zero number may or may not be a significant figure in case of that number does not have a decimal point (500 may have 1, 2 or 3 significant figures).  In this cases use scientific notation to avoid ambiguity.
• The placement of decimal at the trailing zero is indicates the zero is significant (550. has three significant figures).

Answer to Problem 42E

The answer is $\text{7}\text{.82}\text{×}{\text{10}}^{\text{-17}}$.

Explanation of Solution

Given data:

$\raisebox{1ex}{$\text{6}\text{.6262}\text{×}{\text{10}}^{\text{-34}}\text{×}\text{2}\text{.998}\text{×}{\text{10}}^{\text{8}}$}\!\left/ \!\raisebox{-1ex}{$\text{2}\text{.54}\text{×}{\text{10}}^{\text{-9}}$}\right.$

First, take the multiplication of the given values

$\begin{array}{l}\text{6}\text{.6262}\text{×}{\text{10}}^{\text{-34}}\text{×}\text{2}\text{.998}\text{×}{\text{10}}^{\text{8}}\\ \text{=}\text{6}\text{.6262}\text{×}\text{2}\text{.998}\text{×}{\text{10}}^{\text{-26}}\\ \text{=}\text{19}\text{.8653}\text{×}{\text{10}}^{\text{-26}}\end{array}$

Now, take the division of the obtained values

$\begin{array}{l}\frac{\text{19}\text{.8653}\text{×}{\text{10}}^{\text{-26}}}{\text{2}\text{.54}\text{×}{\text{10}}^{\text{-9}}}\text{=}\text{7}\text{.821}\text{×}{\text{10}}^{\text{-26}\text{+}\text{9}}\\ \text{=}\text{7}\text{.821}\text{×}{\text{10}}^{\text{-17}}\end{array}$

The answer is $\text{7}\text{.82}\text{×}{\text{10}}^{\text{-17}}$.

The result is $\text{7}\text{.821}\text{×}{\text{10}}^{\text{-17}}$.  Here the limiting term $\text{2}\text{.54}$ has two decimal places.  Hence, the obtained result was rounded off to get the final result with appropriate number of significant figures.

Interpretation Introduction

Interpretation:

For the given set of mathematical calculations the answer has to be given in appropriate number of significant figures.

Concept introduction:

Significant figures: The digits having a meaning in a given number are called as significant figures.

Significant figures – calculation rules

• Significant figure is a non-zero number (4443.2 has five significant figures)
• Zeros placement in between non-zero number are significant figures (40005 has five significant figures).
• Zeros before the non-zero number are not significant number (0.00040005 has five significant figures).
• Zeros after the non-zero number are significant in case of that number contains decimal point (4.5000 has five significant figures).
• Zeros after the non-zero number may or may not be a significant figure in case of that number does not have a decimal point (500 may have 1, 2 or 3 significant figures).  In this cases use scientific notation to avoid ambiguity.
• The placement of decimal at the trailing zero is indicates the zero is significant (550. has three significant figures).

Answer to Problem 42E

The answer is $\text{0}\text{.20}$.

Explanation of Solution

Given data:

$\text{1}\text{.285}\text{×}{\text{10}}^{\text{-2}}\text{+}\text{1}\text{.24}\text{×}{\text{10}}^{\text{-3}}\text{+}\text{1}\text{.879}\text{×}{\text{10}}^{\text{-1}}$

First, convert the given numbers into same scientific notation

$\begin{array}{l}\text{1}\text{.285}\text{×}{\text{10}}^{\text{-2}}\text{=}\text{1}\text{.285}\text{×}{\text{10}}^{\text{-2}}\\ \text{1}\text{.24}\text{×}{\text{10}}^{\text{-3}}\text{=}\text{0}\text{.124}\text{×}{\text{10}}^{\text{-2}}\\ \text{1}\text{.879×}{\text{10}}^{\text{-1}}\text{=}\text{18}\text{.79}\text{×}{\text{10}}^{\text{-2}}\end{array}$

Now, take the addition of the obtained values

$\begin{array}{l}\left(\text{1}\text{.285}\text{+}\text{0}\text{.124}\text{+}\text{18}\text{.79}\right)\times {10}^{-2}\text{=}\left(\text{20}\text{.199}\right)\times {10}^{-2}\\ =0.20199\end{array}$

The answer is $\text{0}\text{.20}$.

The result is $\text{0}\text{.20199}$.  Here the limiting term $\text{1}\text{.24}$ has two decimal places.  Hence, the obtained result was rounded off to get the final result with appropriate number of significant figures.

Interpretation Introduction

Interpretation:

For the given set of mathematical calculations the answer has to be given in appropriate number of significant figures.

Concept introduction:

Significant figures: The digits having a meaning in a given number are called as significant figures.

Significant figures – calculation rules

• Significant figure is a non-zero number (4443.2 has five significant figures)
• Zeros placement in between non-zero number are significant figures (40005 has five significant figures).
• Zeros before the non-zero number are not significant number (0.00040005 has five significant figures).
• Zeros after the non-zero number are significant in case of that number contains decimal point (4.5000 has five significant figures).
• Zeros after the non-zero number may or may not be a significant figure in case of that number does not have a decimal point (500 may have 1, 2 or 3 significant figures).  In this cases use scientific notation to avoid ambiguity.
• The placement of decimal at the trailing zero is indicates the zero is significant (550. has three significant figures).

Answer to Problem 42E

The answer is $\text{2}\text{.29158}\text{×}{\text{10}}^{\text{-27}}$.

Explanation of Solution

Given data:

$\raisebox{1ex}{$\left(\text{1}\text{.00866}\text{-}\text{1}\text{.00728}\right)$}\!\left/ \!\raisebox{-1ex}{$\text{6}\text{.02205}\text{×}{\text{10}}^{\text{23}}$}\right.$

First, take subtraction of the given values

$\text{1}\text{.00866}\text{-}\text{1}\text{.00728}\text{=}\text{0}\text{.00138}$

Now, take the division of the obtained values

$\begin{array}{l}\frac{\text{0}\text{.00138}}{\text{6}\text{.02205}\text{×}{\text{10}}^{\text{23}}}\text{=}\text{0}\text{.0002291578}\text{×}{\text{10}}^{\text{-23}}\\ \text{=2}\text{.29158}\text{×}{\text{10}}^{\text{-27}}\end{array}$

The answer is $\text{2}\text{.29158}\text{×}{\text{10}}^{\text{-27}}$.

The result is $\text{2}\text{.291578}\text{×}{\text{10}}^{\text{-27}}$.  Here the limiting term $\text{1}\text{.00866}$ has five decimal places.  Hence, the obtained result was rounded off to get the final result with appropriate number of significant figures.

Interpretation Introduction

Interpretation:

For the given set of mathematical calculations the answer has to be given in appropriate number of significant figures.

Concept introduction:

Significant figures: The digits having a meaning in a given number are called as significant figures.

Significant figures – calculation rules

• Significant figure is a non-zero number (4443.2 has five significant figures)
• Zeros placement in between non-zero number are significant figures (40005 has five significant figures).
• Zeros before the non-zero number are not significant number (0.00040005 has five significant figures).
• Zeros after the non-zero number are significant in case of that number contains decimal point (4.5000 has five significant figures).
• Zeros after the non-zero number may or may not be a significant figure in case of that number does not have a decimal point (500 may have 1, 2 or 3 significant figures).  In this cases use scientific notation to avoid ambiguity.
• The placement of decimal at the trailing zero is indicates the zero is significant (550. has three significant figures).

Answer to Problem 42E

The answer is $\text{81}\text{.013}$.

Explanation of Solution

Given data:

$\frac{\text{9}\text{.875}\text{×}{\text{10}}^{\text{2}}\text{-}\text{9}\text{.795}\text{×}{\text{10}}^{\text{2}}}{\text{9}\text{.875}\text{×}{\text{10}}^{\text{2}}}\text{×}\text{100}$

First, take subtraction of the given values

$\text{9}\text{.875}\text{×}{\text{10}}^{\text{2}}\text{-}\text{9}\text{.795}\text{×}{\text{10}}^{\text{2}}\text{=}\text{8}\text{×}{\text{10}}^{\text{2}}$

Now, take the division of the obtained values

$\begin{array}{l}\frac{\text{8}\text{×}{\text{10}}^{\text{2}}}{\text{9}\text{.87}\text{×}{\text{10}}^{\text{2}}}\text{×}\text{100}\text{=}\text{0}\text{.810}\text{×}\text{100}\\ \text{=}\text{81}\text{.01266}\end{array}$

The answer is $\text{81}\text{.013}$.

The result is $\text{81}\text{.01266}$.  Here the limiting terms $\text{9}\text{.875}$ has three decimal places.  Hence, the obtained result was rounded off to get the final result with appropriate number of significant figures.

Interpretation Introduction

Interpretation:

For the given set of mathematical calculations the answer has to be given in appropriate number of significant figures.

Concept introduction:

Significant figures: The digits having a meaning in a given number are called as significant figures.

Significant figures – calculation rules

• Significant figure is a non-zero number (4443.2 has five significant figures)
• Zeros placement in between non-zero number are significant figures (40005 has five significant figures).
• Zeros before the non-zero number are not significant number (0.00040005 has five significant figures).
• Zeros after the non-zero number are significant in case of that number contains decimal point (4.5000 has five significant figures).
• Zeros after the non-zero number may or may not be a significant figure in case of that number does not have a decimal point (500 may have 1, 2 or 3 significant figures).  In this cases use scientific notation to avoid ambiguity.
• The placement of decimal at the trailing zero is indicates the zero is significant (550. has three significant figures).

Answer to Problem 42E

The answer is $\text{11}\text{.30}\text{×}{\text{10}}^{\text{2}}$.

Explanation of Solution

Given data:

$\frac{\text{9}\text{.42}\text{×}{\text{10}}^{\text{2}}\text{+}\text{8}\text{.234}\text{×}{\text{10}}^{\text{2}}\text{+}\text{1}\text{.625}\text{×}{\text{10}}^{\text{3}}}{\text{3}}$

First, convert the given numbers into same scientific notation

$\begin{array}{l}\text{9}\text{.42}\text{×}{\text{10}}^{\text{2}}\text{=}\text{9}\text{.42}\text{×}{\text{10}}^{\text{2}}\\ \text{8}\text{.234}\text{×}{\text{10}}^{\text{2}}\text{=}\text{8}\text{.234}\text{×}{\text{10}}^{\text{2}}\\ \text{1}\text{.625}\text{×}{\text{10}}^{\text{3}}\text{=}\text{16}\text{.25}\text{×}{\text{10}}^{\text{2}}\end{array}$

Then, take the addition of the obtained values

$\left(\text{9}\text{.42}\text{+}\text{8}\text{.234}\text{+}\text{16}\text{.25}\right)\text{×}{\text{10}}^{\text{2}}\text{=}\text{33}\text{.904}\text{×}{\text{10}}^{\text{2}}$

Now take the division of the obtained values

$\frac{\text{33}\text{.904}}{\text{3}}\text{×}{\text{10}}^{\text{2}}\text{=}\text{11}\text{.3013}\text{×}{\text{10}}^{\text{2}}$

The answer is $\text{11}\text{.30}\text{×}{\text{10}}^{\text{2}}$.

The result is $11.3013\text{×}{\text{10}}^{\text{2}}$.  Here the limiting terms $\text{9}\text{.42}$ has two decimal places.  Hence, the obtained result was rounded off to get the final result with appropriate number of significant figures.