Chapter 1, Problem 1.56E

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 nonexponential 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 1.56E

The given multiplication expressed using scientific notation is shown below.

$\left(1.44\times {10}^2\right)\left(8.76\times {10}^{-2}\right)$

The answer of the given multiplication expressed by using scientific notation is $1.26144\times {10}^1$.

Explanation of Solution

The given question is,

$\left(144\right)\left(0.0876\right)$

To express the number $144$ using the scientific notation, the non-zero digits should be separated by decimal andonly one non-zero digit should be written before decimal. Thus, the number can be expressed as $1.44\times {10}^2$.

To express the number $0.0876$ 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 $8.76\times {10}^{-2}$.

The given multiplication expressed using scientific notation is shown below.

$\left(1.44\times {10}^2\right)\left(8.76\times {10}^{-2}\right)$

The multiplicationscan 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 a given multiplication can be calculated as follows:

$\begin{array}{r}\left(1.44\times 8.76\right)\left({10}^{-2+2}\right)\\ & =& 12.6144\times {10}^0\end{array}$

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 $0$ to $1$. Hence, the answer of a given multiplication is $1.26144\times {10}^1$.

Conclusion

The given multiplication expressed using scientific notation is shown below.

$\left(1.44\times {10}^2\right)\left(8.76\times {10}^{-2}\right)$

The answer of the given multiplication expressed by using scientific notation is $1.26144\times {10}^1$.

Interpretation Introduction

(b)

Interpretation:

The given numbers are to be expressed using scientific notation and also the answer of the given multiplicationis 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 nonexponential 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 1.56E

The given multiplication expressed using scientific notation is shown below.

$\left(7.51\times {10}^2\right)\left(1.06\times {10}^2\right)$

The answer of the given multiplication expressed by using scientific notation is $7.9606\times {10}^4$.

Explanation of Solution

The given question is,

$\left(751\right)\left(106\right)$

To express the number $751$ 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 $7.51\times {10}^2$.

To express the number $106$ using the scientific notation, the non-zero digits should be separated by decimal andonly one non-zero digit should be written before decimal. Thus, the number can be expressed as $1.06\times {10}^2$.

The given multiplication expressed using scientific notation is shown below.

$\left(7.51\times {10}^2\right)\left(1.06\times {10}^2\right)$

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 a given multiplication can be calculated as follows:

$\begin{array}{r}\left(7.51\times 1.06\right)\left({10}^{2+2}\right)\\ & =& 7.9606\times {10}^4\end{array}$

Hence, the answer of a given multiplication is $7.9606\times {10}^4$.

Conclusion

The given multiplication expressed using scientific notation is shown below.

$\left(7.51\times {10}^2\right)\left(1.06\times {10}^2\right)$

The answer of the given multiplication expressed by using scientific notation is $7.9606\times {10}^4$.

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 nonexponential 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 1.56E

The given multiplication expressed using scientific notation is shown below.

$\left(4.22\times {10}^{-2}\right)\left(1.19\times {10}^{-3}\right)$

The answer of the given multiplication expressed by using scientific notation is $5.0218\times {10}^{-5}$.

Explanation of Solution

The given question is,

$\left(0.0422\right)\left(0.00119\right)$

To express the number $0.0422$ 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 $4.22\times {10}^{-2}$.

To express the number $0.00119$ using the scientific notation, decimal moves three 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 $1.19\times {10}^{-3}$.

The given multiplication expressed using scientific notation is shown below.

$\left(4.22\times {10}^{-2}\right)\left(1.19\times {10}^{-3}\right)$

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 a given multiplication can be calculated as follows:

$\begin{array}{r}\left(4.22\times 1.19\right)\left({10}^{-2-3}\right)\\ & =& 5.0218\times {10}^{-5}\end{array}$

Hence, the answer of a given multiplication is $5.0218\times {10}^{-5}$.

Conclusion

The given multiplication expressed using scientific notation is shown below.

$\left(4.22\times {10}^{-2}\right)\left(1.19\times {10}^{-3}\right)$

The answer of the given multiplication expressed by using scientific notation is $5.0218\times {10}^{-5}$.

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 nonexponential 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 1.56E

The given multiplication expressed using scientific notation is shown below.

$\left(1.28\times {10}^5\right)\left(3.16\times {10}^{-5}\right)$

The answer of the given multiplication expressed by using scientific notation is $4.0448\times {10}^0$.

Explanation of Solution

The given question is,

$\left(128,000\right)\left(0.0000316\right)$

To express the number $128,000$ 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 $1.28\times {10}^5$.

To express the number $0.0000316$ using the scientific notation, decimal moves five 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 $3.16\times {10}^{-5}$.

The given multiplication expressed using scientific notation is shown below.

$\left(1.28\times {10}^5\right)\left(3.16\times {10}^{-5}\right)$

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 a given multiplication can be calculated as follows:

$\begin{array}{r}\left(1.28\times 3.16\right)\left({10}^{5-5}\right)\\ & =& 4.0448\times {10}^0\end{array}$

Hence, the answer of a given multiplication is $4.0448\times {10}^0$.

Conclusion

The given multiplication expressed using scientific notation is shown below.

$\left(1.28\times {10}^5\right)\left(3.16\times {10}^{-5}\right)$

The answer of the given multiplication expressed by using scientific notation is $4.0448\times {10}^0$.