Chapter 1, Problem 2P

(a)

To determine

The scientific notation forms of the numbers $1000;0.000001;$ $1001;1,000,000,000,000,000$; $123,000;$ $0.000456$.

Explanation of Solution

Formula Used:

A number in scientific notation is in form $a\times {10}^n$, where $1\le \left|a\right|<10$ and $n$ is integer.

Explanation:

Consider the number $1000$. It is greater than $10$. So, positive power of $10$ is used. Then, the number $1000$ in scientific notation, that is in the form $a\times {10}^n$, will be $1\times {10}^3$.

Now, consider the number $0.000001$. The number is in between $0$ and $1$. So, negative power of $10$ is used in the scientific notation. Then, the number $0.000001$ in scientific notation will be $1\times {10}^{-6}$.

Similarly, the number $1001$ will be written in scientific notation as $1.001\times {10}^3$, the number $1,000,000,000,000,000$ as $1.00\times {10}^{15}$, and the number $123,000$ as $1.23\times {10}^5$, since the number is greater than $10$.

But, the number $0.000456$ will be written in scientific notation as $4.56\times {10}^{-4}$ since the number is between $0$ and $1$.

(b)

To determine

To write: The numbers $3.16\times {10}^7;2.998\times {10}^5;$ $6.67\times {10}^{-11};\text{ and 2}\times {10}^0$ in the “normal” numerical form.

Explanation of Solution

Formula Used:

A normal numerical form is the one in which the number is not written in the exponent form.

Explanation:

Consider the number $3.16\times {10}^7$. In the “normal” numerical form, it will be written as,

$\begin{array}{c}3.16\times {10}^7=3.16\times 10,000,000\\ =\frac{316}{100}\times 10,000,000\\ =31,600,000\end{array}$

Similarly, consider the number $2.998\times {10}^5$. In the “normal” numerical form, it will be written as,

$\begin{array}{c}2.998\times {10}^5=2.998\times 100,000\\ =\frac{2998}{1000}\times 100,000\\ =299,800\end{array}$

And, the number $6.67\times {10}^{-11}$ in the “normal” numerical form will be,

$\begin{array}{c}6.67\times {10}^{-11}=\frac{6.67}{100,000,000,000}\\ =0.0000000000667\end{array}$

And, the number $2\times {10}^0$ in the “normal” numerical form will be,

$\begin{array}{c}2\times {10}^0=2\times 1\\ =2\end{array}$

(c)

To determine

The value of the expressions $\left(2\times {10}^3\right)+{10}^{-2}$;$\left(1.99\times {10}^{30}\right)/\left(5.98\times {10}^{24}\right)$, and $\left(3.16\times {10}^7\right)\times \left(2.998\times {10}^5\right)$.

Explanation of Solution

Formula Used:

Rule for the multiplication of the numbers with exponents: Multiply the factors and add the exponents.

Rule for the division of the numbers with exponents: Divide the factors and subtract the exponents.

Explanation:

Consider the expression $\left(2\times {10}^3\right)+{10}^{-2}$.

For adding the two numbers, both should have same exponents. Here, the exponents are not same thus converting both the numbers in normal numerical form and then adding,

$\begin{array}{c}\left(2\times {10}^3\right)+{10}^{-2}=2000+0.01\\ =2000.01\end{array}$

Now, consider the expression $\left(1.99\times {10}^{30}\right)/\left(5.98\times {10}^{24}\right)$.

For dividing the two numbers, divide the factors and subtract the exponents.

$\begin{array}{c}\left(1.99\times {10}^{30}\right)/\left(5.98\times {10}^{24}\right)=\left(\frac{1.99}{5.98}\right)\times {10}^{30-24}\\ =0.332\times {10}^6\end{array}$

Consider the expression $\left(3.16\times {10}^7\right)\times \left(2.998\times {10}^5\right)$.

For multiplying the two numbers, multiply the factors and add the exponents.

$\begin{array}{c}\left(3.16\times {10}^7\right)\times \left(2.998\times {10}^5\right)=\left(3.16\times 2.998\right)\times {10}^{7+5}\\ =9.47\times {10}^{62}\end{array}$