Chapter 2, Problem 99A

(a)

Interpretation Introduction

Interpretation:

The answer for the given multiplication problem needs to be written in scientific notation and round off to the correct significant figures.

$\left(5.31\times {10}^{-2}\text{cm}\right)\times \left(2.46\times {10}^5\text{cm}\right)$

Concept introduction:

Scientific notation is a term used to express any number as a number between $1$ and $10$ (termed as coefficient) multiplied by $10$ raised to a power (termed as exponent). Scientists express numbers in a scientific notation form.

Answer to Problem 99A

The complete multiplication of the given problem is $1.31\times {10}^4{\text{cm}}^2$.

Explanation of Solution

When multiplying the numbers in scientific notation, add the exponent values. Simply, multiply both numbers and add the exponent values.

$\begin{array}{c}\left(5.31\times {10}^{-2}\text{cm}\right)\times \left(2.46\times {10}^5\text{cm}\right)=\left(5.31\times 2.46\right)\times {10}^{-2+5}{\text{cm}}^2\\ \text{=}\text{13}\text{.0626}\times {10}^3{\text{cm}}^2\\ =1.31\times {10}^4{\text{cm}}^2\end{array}$

When multiplying the numbers, answer must have the same number of significant figure as the measurement with lowest number of significant figures. Both the given numbers in multiplication have the same number of significant figures i.e. three. Therefore, the solution must contain three significant figures. Hence, the answer is $1.31\times {10}^4{\text{cm}}^2$.

(b)

Interpretation Introduction

Interpretation:

The answer for the given multiplication problem needs to be written in scientific notation and round off to the correct significant figures.

$\left(3.78\times {10}^3\text{m}\right)\times \left(7.21\times {10}^2\text{m}\right)$

Concept introduction:

Scientific notation is a term used to express any number as a number between $1$ and $10$ (termed as coefficient) multiplied by $10$ raised to a power (termed as exponent). Scientists express numbers in a scientific notation form.

Answer to Problem 99A

The complete multiplication of the given problem is $2.73\times {10}^6{\text{m}}^2$.

Explanation of Solution

When multiplying the numbers in scientific notation, add the exponent values. Simply, multiply both numbers and add the exponent values.

$\begin{array}{c}\left(3.78\times {10}^3\text{m}\right)\times \left(7.21\times {10}^2\text{m}\right)=\left(3.78\times 7.21\right)\times {10}^{3+2}{\text{m}}^2\\ \text{=}\text{27}\text{.2538}\times {10}^5{\text{m}}^2\\ =2.73\times {10}^6{\text{m}}^2\end{array}$

When multiplying the numbers, answer must have the same number of significant figure as the measurement with lowest number of significant figures. Both the given numbers in multiplication have the same number of significant figures i.e. three. Therefore, the solution must contain three significant figures. Hence, the answer is $2.73\times {10}^6{\text{m}}^2$.

(c)

Interpretation Introduction

Interpretation:

The answer for the given multiplication problem needs to be written in scientific notation and round off to the correct significant figures.

$\left(8.12\times {10}^{-3}\text{m}\right)\times \left(1.14\times {10}^{-5}\text{m}\right)$

Concept introduction:

Scientific notation is a term used to express any number as a number between $1$ and $10$ (termed as coefficient) multiplied by $10$ raised to a power (termed as exponent). Scientists express numbers in a scientific notation form.

Answer to Problem 99A

The complete multiplication of the given problem is $9.26\times {10}^{-8}{\text{m}}^2$.

Explanation of Solution

When multiplying the numbers in scientific notation, add the exponent values. Simply, multiply both numbers and add the exponent values.

$\begin{array}{c}\left(8.12\times {10}^{-3}\text{m}\right)\times \left(1.14\times {10}^{-5}\text{m}\right)=\left(8.12\times 1.14\right)\times {10}^{-3-5}{\text{m}}^2\\ \text{=}\text{9}\text{.2568}\times {10}^{-8}{\text{m}}^2\\ =9.26\times {10}^{-8}{\text{m}}^2\end{array}$

When multiplying the numbers, answer must have the same number of significant figure as the measurement with lowest number of significant figures. Both the given numbers in multiplication have the same number of significant figures i.e. three. Therefore, the solution must contain three significant figures. Hence, the answer is $9.26\times {10}^{-8}{\text{m}}^2$.

(d)

Interpretation Introduction

Interpretation:

The answer for the given multiplication problem needs to be written in scientific notation and round off to the correct significant figures.

$\left(9.33\times {10}^4\text{mm}\right)÷\left(3.0\times {10}^2\text{mm}\right)$

Concept introduction:

Scientific notation is a term used to express any number as a number between $1$ and $10$ (termed as coefficient) multiplied by $10$ raised to a power (termed as exponent). Scientists express numbers in a scientific notation form.

Answer to Problem 99A

The complete division of the given problem is $3.\text{1}\times {10}^2$.

Explanation of Solution

When dividing the numbers in scientific notation, subtract the exponent values. Simply, divide both numbers and subtract the exponent values.

$\begin{array}{c}\left(9.33\times {10}^4\text{mm}\right)÷\left(3.0\times {10}^2\text{mm}\right)=\left(9.33÷3.0\right)\times {10}^{4-2}\\ \text{=}\text{3}\text{.11}\times {10}^2\\ =3.\text{1}\times {10}^2\end{array}$

When multiplying the numbers, answer must have the same number of significant figure as the measurement with lowest number of significant figures. The number with lowest significant figure i.e. two is $3.0\times {10}^2\text{mm}$. Therefore, the solution must contain two significant figures. Hence, the answer is $3.\text{1}\times {10}^2$.

(e)

Interpretation Introduction

Interpretation:

The answer for the given multiplication problem needs to be written in scientific notation and round off to the correct significant figures.

$\left(4.42\times {10}^{-3}\text{kg}\right)÷\left(2.0\times {10}^2\text{kg}\right)$

Concept introduction:

Scientific notation is a term used to express any number as a number between $1$ and $10$ (termed as coefficient) multiplied by $10$ raised to a power (termed as exponent). Scientists express numbers in a scientific notation form.

Answer to Problem 99A

The complete division of the given problem is $2.2\times {10}^{-5}$.

Explanation of Solution

When dividing the numbers in scientific notation, subtract the exponent values. Simply, divide both numbers and subtract the exponent values.

$\begin{array}{c}\left(4.42\times {10}^{-3}\text{kg}\right)÷\left(2.0\times {10}^2\text{kg}\right)=\left(4.42÷2.0\right)\times {10}^{-3-2}\\ \text{=}2.\text{21}\times {10}^{-5}\\ =2.2\times {10}^{-5}\end{array}$

When multiplying the numbers, answer must have the same number of significant figure as the measurement with lowest number of significant figures. The number with lowest significant figure i.e. two is $2.0\times {10}^2\text{kg}$. Therefore, the solution must contain two significant figures. Hence, the answer is $2.2\times {10}^{-5}$.

(f)

Interpretation Introduction

Interpretation:

The answer for the given multiplication problem needs to be written in scientific notation and round off to the correct significant figures.

$\left(6.42\times {10}^{-2}\text{g}\right)÷\left(3.21\times {10}^{-3}\text{g}\right)$

Concept introduction:

Scientific notation is a term used to express any number as a number between $1$ and $10$ (termed as coefficient) multiplied by $10$ raised to a power (termed as exponent). Scientists express numbers in a scientific notation form.

Answer to Problem 99A

The complete division of the given problem is $2.00\times {10}^1$.

Explanation of Solution

When dividing the numbers in scientific notation, subtract the exponent values. Simply, divide both numbers and subtract the exponent values.

$\begin{array}{c}\left(6.42\times {10}^{-2}\text{g}\right)÷\left(3.21\times {10}^{-3}\text{g}\right)=\left(6.42÷3.21\right)\times {10}^{-2-\left(-3\right)}\\ \text{=}2.\text{00}\times {10}^{-2+3}\\ =2.00\times {10}^1\end{array}$

When multiplying the numbers, answer must have the same number of significant figure as the measurement with lowest number of significant figures. Both the given numbers in multiplication have the same number of significant figures i.e. three. Therefore, the solution must contain two significant figures. Hence, the answer is $2.00\times {10}^1$.