Chapter 2, Problem 2.66EP

(a)

Interpretation Introduction

Interpretation:

Answer for $\left(\text{4}\text{.0}\text{x}{\text{10}}^4\right)\text{x}\left(\text{1}\text{.32}\text{x}{\text{10}}^8\right)$ has to be expressed in scientific notation with the correct number of significant figures.

Concept Introduction:

In scientific work, large and very small numbers occurs frequently.  But it is been observed that, to record these vast numbers or very small numbers is difficult because, the numbers may get missed while recording them.  It is time consuming and the possibilities of error occurrence are also high.  Hence to overcome these demerits, a method called scientific notation is used.  Scientific notation is a numerical system where a decimal number is expressed as product of two number between 1 and 10 (coefficient) and 10 that is raised to power (exponential term).  In this method the numbers are expressed in form of “${\text{A x 10}}^{\text{n}}$, where A is called the co-efficient, n is the whole number and 10 is called the exponential term.  This way of expressing is much easier and error free.  If the decimal number is 10 or greater than 10, then the exponent is positive and if the decimal number is less than 1, then the exponent is negative.

For multiplication of the scientific notation, the coefficients are multiplied as such and the exponents are added.  For division of the scientific notation, the coefficients are divided as such and the exponents are subtracted.

Answer to Problem 2.66EP

Answer for $\left(\text{4}\text{.0}\text{x}{\text{10}}^4\right)\text{x}\left(\text{1}\text{.32}\text{x}{\text{10}}^8\right)$ is $\text{5}\text{.3}\text{x}{\text{10}}^{\text{12}}$.

Explanation of Solution

Given scientific notation is $\left(\text{4}\text{.0}\text{x}{\text{10}}^4\right)\text{x}\left(\text{1}\text{.32}\text{x}{\text{10}}^8\right)$.

The coefficient has to be multiplied and the exponents has to be added.  This can be done as shown below,

  $\begin{array}{l}\left(\text{4}\text{.0}\text{x}{\text{10}}^4\right)\text{x}\left(\text{1}\text{.32}\text{x}{\text{10}}^8\right)\text{=}\text{5}\text{.28}\text{x}{\text{10}}^{\text{4+8}}\\ \text{=}\text{5}\text{.28}\text{x}{\text{10}}^{\text{12}}\end{array}$

This value has to be expressed to the correct number of significant figures.  One of the number that is multiplied has three significant figures while the other has two significant figures.  Hence, the answer is expressed with two significant figures as $\text{5}\text{.3}\text{x}{\text{10}}^{\text{12}}$.

(b)

Interpretation Introduction

Interpretation:

Answer for $\left(\text{3}\text{.200}\text{x}{\text{10}}^{\text{7}}\right)\text{x}\left(\text{1}\text{.10}\text{x}{\text{10}}^{\text{-2}}\right)\text{x}\left(\text{2}\text{.3}\text{x}{\text{10}}^{\text{-7}}\right)$ has to be expressed in scientific notation with the correct number of significant figures.

Concept Introduction:

In scientific work, large and very small numbers occurs frequently.  But it is been observed that, to record these vast numbers or very small numbers is difficult because, the numbers may get missed while recording them.  It is time consuming and the possibilities of error occurrence are also high.  Hence to overcome these demerits, a method called scientific notation is used.  Scientific notation is a numerical system where a decimal number is expressed as product of two number between 1 and 10 (coefficient) and 10 that is raised to power (exponential term).  In this method the numbers are expressed in form of “${\text{A x 10}}^{\text{n}}$, where A is called the co-efficient, n is the whole number and 10 is called the exponential term.  This way of expressing is much easier and error free.  If the decimal number is 10 or greater than 10, then the exponent is positive and if the decimal number is less than 1, then the exponent is negative.

For multiplication of the scientific notation, the coefficients are multiplied as such and the exponents are added.  For division of the scientific notation, the coefficients are divided as such and the exponents are subtracted.

Answer to Problem 2.66EP

Answer for $\left(\text{3}\text{.200}\text{x}{\text{10}}^{\text{7}}\right)\text{x}\left(\text{1}\text{.10}\text{x}{\text{10}}^{\text{-2}}\right)\text{x}\left(\text{2}\text{.3}\text{x}{\text{10}}^{\text{-7}}\right)$ is $\text{8}\text{.1}\text{x}{\text{10}}^{\text{-2}}$.

Explanation of Solution

Given scientific notation is $\left(\text{3}\text{.200}\text{x}{\text{10}}^{\text{7}}\right)\text{x}\left(\text{1}\text{.10}\text{x}{\text{10}}^{\text{-2}}\right)\text{x}\left(\text{2}\text{.3}\text{x}{\text{10}}^{\text{-7}}\right)$.

The coefficient has to be multiplied and the exponents has to be added.  This can be done as shown below,

  $\begin{array}{l}\left(\text{3}\text{.200}\text{x}{\text{10}}^{\text{7}}\right)\text{x}\left(\text{1}\text{.10}\text{x}{\text{10}}^{\text{-2}}\right)\text{x}\left(\text{2}\text{.3}\text{x}{\text{10}}^{\text{-7}}\right)\text{=}\text{8}\text{.096}\text{x}{\text{10}}^{\text{7-2-7}}\\ \text{=}\text{8}\text{.096}\text{x}{\text{10}}^{-2}\end{array}$

This value has to be expressed to the correct number of significant figures.  One of the number that is multiplied has two significant figures.  Hence, the answer is expressed with two significant figures as $\text{8}\text{.1}\text{x}{\text{10}}^{-2}$.

(c)

Interpretation Introduction

Interpretation:

Answer for $\left(6.0\text{x}{\text{10}}^{\text{-5}}\right)÷\left(\text{3}\text{.0}\text{x}{\text{10}}^3\right)$ has to be expressed in scientific notation with the correct number of significant figures.

Concept Introduction:

In scientific work, large and very small numbers occurs frequently.  But it is been observed that, to record these vast numbers or very small numbers is difficult because, the numbers may get missed while recording them.  It is time consuming and the possibilities of error occurrence are also high.  Hence to overcome these demerits, a method called scientific notation is used.  Scientific notation is a numerical system where a decimal number is expressed as product of two number between 1 and 10 (coefficient) and 10 that is raised to power (exponential term).  In this method the numbers are expressed in form of “${\text{A x 10}}^{\text{n}}$, where A is called the co-efficient, n is the whole number and 10 is called the exponential term.  This way of expressing is much easier and error free.  If the decimal number is 10 or greater than 10, then the exponent is positive and if the decimal number is less than 1, then the exponent is negative.

For multiplication of the scientific notation, the coefficients are multiplied as such and the exponents are added.  For division of the scientific notation, the coefficients are divided as such and the exponents are subtracted.

Answer to Problem 2.66EP

Answer for $\left(6.0\text{x}{\text{10}}^{\text{-5}}\right)÷\left(\text{3}\text{.0}\text{x}{\text{10}}^3\right)$ is $\text{2}\text{.0}\text{x}{\text{10}}^{\text{-8}}$.

Explanation of Solution

Given scientific notation is $\left(6.0\text{x}{\text{10}}^{\text{-5}}\right)÷\left(\text{3}\text{.0}\text{x}{\text{10}}^3\right)$.

The coefficient has to be divided and the exponents has to be subtracted.  This can be done as shown below,

  $\begin{array}{l}\left(6.0\text{x}{\text{10}}^{\text{-5}}\right)÷\left(\text{3}\text{.0}\text{x}{\text{10}}^3\right)\text{=}\text{2}\text{.0}\text{x}{\text{10}}^{\text{-5-3}}\\ \text{=}\text{2}\text{.0}\text{x}{\text{10}}^{-8}\end{array}$

This value has to be expressed to the correct number of significant figures.  Both the scientific notation has got two significant figures.  Hence, the answer is expressed with two significant figures as $\text{2}\text{.0}\text{x}{\text{10}}^{-8}$.

(d)

Interpretation Introduction

Interpretation:

Answer for $\left(\text{3}\text{.2}\text{x}{\text{10}}^{\text{2}}\right)\text{x}\left(\text{3}\text{.31}\text{x}{\text{10}}^6\right)/\left(4.00\text{x}{\text{10}}^{\text{-3}}\right)\text{x}\left(\text{2}\text{.0}\text{x}{\text{10}}^6\right)$ has to be expressed in scientific notation with the correct number of significant figures.

Concept Introduction:

In scientific work, large and very small numbers occurs frequently.  But it is been observed that, to record these vast numbers or very small numbers is difficult because, the numbers may get missed while recording them.  It is time consuming and the possibilities of error occurrence are also high.  Hence to overcome these demerits, a method called scientific notation is used.  Scientific notation is a numerical system where a decimal number is expressed as product of two number between 1 and 10 (coefficient) and 10 that is raised to power (exponential term).  In this method the numbers are expressed in form of “${\text{A x 10}}^{\text{n}}$, where A is called the co-efficient, n is the whole number and 10 is called the exponential term.  This way of expressing is much easier and error free.  If the decimal number is 10 or greater than 10, then the exponent is positive and if the decimal number is less than 1, then the exponent is negative.

For multiplication of the scientific notation, the coefficients are multiplied as such and the exponents are added.  For division of the scientific notation, the coefficients are divided as such and the exponents are subtracted.

Answer to Problem 2.66EP

Answer for $\left(\text{3}\text{.2}\text{x}{\text{10}}^{\text{2}}\right)\text{x}\left(\text{3}\text{.31}\text{x}{\text{10}}^6\right)/\left(4.00\text{x}{\text{10}}^{\text{-3}}\right)\text{x}\left(\text{2}\text{.0}\text{x}{\text{10}}^6\right)$ is $\text{1}\text{.3}\text{x}{\text{10}}^5$.

Explanation of Solution

Given scientific notation is $\left(\text{3}\text{.2}\text{x}{\text{10}}^{\text{2}}\right)\text{x}\left(\text{3}\text{.31}\text{x}{\text{10}}^6\right)/\left(4.00\text{x}{\text{10}}^{\text{-3}}\right)\text{x}\left(\text{2}\text{.0}\text{x}{\text{10}}^6\right)$.

For multiplication, the coefficient has to be multiplied and the exponent has to be added and for division the coefficient has to be divided and the exponents has to be subtracted.  This can be done as shown below,

  $\begin{array}{l}\left(\text{3}\text{.2}\text{x}{\text{10}}^{\text{2}}\right)\text{x}\left(\text{3}\text{.31}\text{x}{\text{10}}^6\right)/\left(4.00\text{x}{\text{10}}^{\text{-3}}\right)\text{x}\left(\text{2}\text{.0}\text{x}{\text{10}}^6\right)\\ \text{=}\left(\text{10}\text{.592}\text{x}{\text{10}}^{\text{2+6}}\right)÷\left(\text{8}\text{.00}\text{x}{\text{10}}^{\text{-3+6}}\right)\\ \text{=}\left(\text{10}\text{.592}\text{x}{\text{10}}^8\right)÷\left(\text{8}\text{.00}\text{x}{\text{10}}^3\right)\\ =\text{1}\text{.324}\text{x}{\text{10}}^{\text{8-3}}\\ =1.324\text{x}{10}^5\end{array}$

This value has to be expressed to the correct number of significant figures.  In scientific notations the least number of significant figure present is two.  Hence, the answer is expressed with two significant figures as $\text{1}\text{.3}\text{x}{\text{10}}^5$.