Chapter 1, Problem 1.57E

Express each of the following numbers using scientific notation, then carry out the multiplication. Express each answer using scientific notation.

a. $\left(538\right)\left(0.154\right)$

b. $\left(600\right)\left(524\right)$

c. $\left(22.8\right)\left(341\right)$

d. $\left(23.6\right)\left(0.047\right)$

Interpretation Introduction

(a)

Interpretation:

The given numbers are to be expressed using scientific notation and also the answer of the given multiplication is to be expressed by using scientific notation.

Concept introduction:

Scientific notation is used to write large numbers or small numbers as the product of a non-exponential term and exponential term in the form of $M\times {10}^n$. The $M$ is any number between $1$ to $10$ written with a decimal after first non-zero digit in the number and $n$ is the exponent which may be positive or negative.

Answer to Problem 1.57E

The given multiplication expressed using scientific notation is shown below.

$\left(5.38\times {10}^2\right)\left(1.54\times {10}^{-1}\right)$

The answer of the given multiplication expressed by using scientific notation is $8.2852\times {10}^1$.

Explanation of Solution

The given question is,

$\left(538\right)\left(0.154\right)$

To express the number $538$ using the scientific notation, the non-zero digits should be separated by decimal and only one non-zero digit should be written before decimal. Thus, the number can be expressed as $5.38\times {10}^2$.

To express the number $0.154$ using the scientific notation, decimal is moved one place to the right and only one non-zero digit should be written before decimal. The number of places that the decimal moves to the right becomes exponent $n$ in the form of $M\times {10}^n$. Thus, the number can be expressed as $1.54\times {10}^{-1}$.

The given multiplication expressed using scientific notation is shown below.

$\left(5.38\times {10}^2\right)\left(1.54\times {10}^{-1}\right)$

The multiplication can be expressed as follows:

$\left(a\times {10}^y\right)\left(b\times {10}^z\right)=\left(a\times b\right)\left({10}^{y+z}\right)$

The answer of the given multiplication can be calculated as follows:

$\begin{array}{r}\left(5.38\times 1.54\right)\left({10}^{2-1}\right)\\ & =& 8.2852\times {10}^1\end{array}$

Hence, the answer of the given multiplication is $8.2852\times {10}^1$.

Conclusion

The given multiplication expressed using scientific notation is shown below.

$\left(5.38\times {10}^2\right)\left(1.54\times {10}^{-1}\right)$

The answer of the given multiplication expressed by using scientific notation is $8.2852\times {10}^1$.

Interpretation Introduction

(b)

Interpretation:

The given numbers are to be expressed using scientific notation and also the answer of the given multiplication is to be expressed by using scientific notation.

Concept introduction:

Scientific notation is used to write large numbers or small numbers as the product of a non-exponential term and exponential term in the form of $M\times {10}^n$. The $M$ is any number between $1$ to $10$ written with a decimal after first non-zero digit in the number and $n$ is the exponent which may be positive or negative.

Answer to Problem 1.57E

The given multiplication expressed using scientific notation is shown below.

$\left(6.00\times {10}^2\right)\left(5.24\times {10}^2\right)$

The answer of the given multiplication expressed by using scientific notation is $3.144\times {10}^5$.

Explanation of Solution

The given question is,

$\left(600\right)\left(524\right)$

To express the number $600$ using the scientific notation, the non-zero digits should be separated by decimal and only one non-zero digit should be written before decimal. Thus, the number can be expressed as $6.00\times {10}^2$.

To express the number $524$ using the scientific notation, the non-zero digits should be separated by decimal and only one non-zero digit should be written before decimal. Thus, the number can be expressed as $5.24\times {10}^2$.

The given multiplication expressed using scientific notation is shown below.

$\left(6.00\times {10}^2\right)\left(5.24\times {10}^2\right)$

The multiplications can be expressed as follows:

$\left(a\times {10}^y\right)\left(b\times {10}^z\right)=\left(a\times b\right)\left({10}^{y+z}\right)$

The answer of the given multiplication can be calculated as follows:

$\begin{array}{r}\left(6.00\times 5.24\right)\left({10}^{2+2}\right)\\ & =& 31.44\times {10}^4\end{array}$

The above answer is adjusted to a standard position. The decimal is shifted one place to the left and the power of $10$ is changed from $4$ to $5$. Hence, the answer of the given multiplication is $3.144\times {10}^5$.

Conclusion

The given multiplication expressed using scientific notation is shown below.

$\left(6.00\times {10}^2\right)\left(5.24\times {10}^2\right)$

The answer of the given multiplication expressed by using scientific notation is $3.144\times {10}^5$.

Interpretation Introduction

(c)

Interpretation:

The given numbers are to be expressed using scientific notation and also the answer of the given multiplication is to be expressed by using scientific notation.

Concept introduction:

Scientific notation is used to write large numbers or small numbers as the product of a non-exponential term and exponential term in the form of $M\times {10}^n$. The $M$ is any number between $1$ to $10$ written with a decimal after first non-zero digit in the number and $n$ is the exponent which may be positive or negative.

Answer to Problem 1.57E

The given multiplication expressed using scientific notation is shown below.

$\left(2.28\times {10}^1\right)\left(3.41\times {10}^2\right)$

The answer of the given multiplication expressed by using scientific notation is $7.7748\times {10}^3$.

Explanation of Solution

The given question is,

$\left(22.8\right)\left(341\right)$

To express the number $22.8$ using the scientific notation, the non-zero digits should be separated by decimal and only one non-zero digit should be written before decimal. Thus, the number can be expressed as $7.7748\times {10}^3$.

To express the number $341$ using the scientific notation, the non-zero digits should be separated by decimal and only one non-zero digit should be written before decimal. Thus, the number can be expressed as $3.41\times {10}^2$.

The given multiplication expressed using scientific notation is shown below.

$\left(2.28\times {10}^1\right)\left(3.41\times {10}^2\right)$

The multiplications can be expressed as follows:

$\left(a\times {10}^y\right)\left(b\times {10}^z\right)=\left(a\times b\right)\left({10}^{y+z}\right)$

The answer of the given multiplication can be calculated as follows:

$\begin{array}{r}\left(2.28\times 3.41\right)\left({10}^{1+2}\right)\\ & =& 7.7748\times {10}^3\end{array}$

Hence, the answer of the given multiplication is $7.7748\times {10}^3$.

Conclusion

The given multiplication expressed using scientific notation is shown below.

$\left(2.28\times {10}^1\right)\left(3.41\times {10}^2\right)$

The answer of the given multiplication expressed by using scientific notation is $7.7748\times {10}^3$.

Interpretation Introduction

(d)

Interpretation:

The given numbers are to be expressed using scientific notation and also the answer of the given multiplication is to be expressed by using scientific notation.

Concept introduction:

Scientific notation is used to write large numbers or small numbers as the product of a non-exponential term and exponential term in the form of $M\times {10}^n$. The $M$ is any number between $1$ to $10$ written with a decimal after first non-zero digit in the number and $n$ is the exponent which may be positive or negative.

Answer to Problem 1.57E

The given multiplication expressed using scientific notation is shown below.

$\left(2.36\times {10}^1\right)\left(4.7\times {10}^{-2}\right)$

The answer of the given multiplication expressed by using scientific notation is $1.1092\times {10}^0$.

Explanation of Solution

The given question is,

$\left(23.6\right)\left(0.047\right)$

To express the number $23.6$ using the scientific notation, the non-zero digits should be separated by decimal and only one non-zero digit should be written before decimal. Thus, the number can be expressed as $2.36\times {10}^1$.

To express the number $0.047$ using the scientific notation, decimal moves two places to the right and only one non-zero digit should be written before decimal. The number of places that the decimal moves to the right becomes exponent $n$ in the form of $M\times {10}^n$. Thus, the number can be expressed as $4.7\times {10}^{-2}$.

The given multiplication expressed using scientific notation is shown below.

$\left(2.36\times {10}^1\right)\left(4.7\times {10}^{-2}\right)$

The multiplications can be expressed as follows:

$\left(a\times {10}^y\right)\left(b\times {10}^z\right)=\left(a\times b\right)\left({10}^{y+z}\right)$

The answer of the given multiplication can be calculated as follows:

$\begin{array}{r}\left(2.36\times 4.7\right)\left({10}^{1-2}\right)\\ & =& 11.092\times {10}^{-1}\end{array}$

The above answer is adjusted to a standard position. The decimal is shifted towards one place left and the power of $10$ is changed from $-1$ to $0$. Hence, the answer of the given multiplication is $1.1092\times {10}^0$.

Conclusion

The given multiplication expressed using scientific notation is shown below.

$\left(2.36\times {10}^1\right)\left(4.7\times {10}^{-2}\right)$

The answer of the given multiplication expressed by using scientific notation is $1.1092\times {10}^0$.