Chapter 1, Problem 1.54E

Do the following multiplications, and express each answer using scientific notation:

a. $\left(8.2\times {10}^{-3}\right)\left(1.1\times {10}^{-2}\right)$

b. $\left(2.7\times {10}^2\right)\left(5.1\times {10}^4\right)$

c. $\left(3.3\times {10}^{-4}\right)\left(2.3\times {10}^2\right)$

d. $\left(9.2\times {10}^{-4}\right)\left(2.1\times {10}^4\right)$

e. $\left(4.3\times {10}^6\right)\left(6.1\times {10}^5\right)$

Interpretation Introduction

(a)

Interpretation:

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 1.54E

The answer of the given multiplication expressed by using scientific notation is $9.02\times {10}^{-5}$.

Explanation of Solution

The given question is,

$\left(8.2\times {10}^{-3}\right)\left(1.1\times {10}^{-2}\right)$

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:

$\begin{array}{r}\left(8.2\times 1.1\right)\left({10}^{-3-2}\right)\\ & =& 9.02\times {10}^{-5}\end{array}$

Hence, the answer of the given multiplication is $9.02\times {10}^{-5}$.

Conclusion

The answer of the given multiplication expressed by using scientific notation is $9.02\times {10}^{-5}$.

Interpretation Introduction

(b)

Interpretation:

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 1.54E

The answer of the given multiplication expressed by using scientific notation is $1.377\times {10}^7$.

Explanation of Solution

The given question is,

$\left(2.7\times {10}^2\right)\left(5.1\times {10}^4\right)$

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:

$\begin{array}{r}\left(2.7\times 5.1\right)\left({10}^{2+4}\right)\\ & =& 13.77\times {10}^6\end{array}$

The above answer is adjusted to a standard position. The decimal is shifted towards one place to the left and the power of $10$ is changed from $6$ to $7$. Hence, the answer of the given multiplication is $1.377\times {10}^7$.

Conclusion

The answer of the given multiplication expressed by using scientific notation is $1.377\times {10}^7$.

Interpretation Introduction

(c)

Interpretation:

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 1.54E

The answer of the given multiplication expressed by using scientific notation is $7.59\times {10}^{-2}$.

Explanation of Solution

The given question is,

$\left(3.3\times {10}^{-4}\right)\left(2.3\times {10}^2\right)$

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:

$\begin{array}{r}\left(3.3\times 2.3\right)\left({10}^{-4+2}\right)\\ & =& 7.59\times {10}^{-2}\end{array}$

Hence, the answer of the given multiplicationis $7.59\times {10}^{-2}$.

Conclusion

The answer of the given multiplication expressed by using scientific notation is $7.59\times {10}^{-2}$.

Interpretation Introduction

(d)

Interpretation:

The answers of the given multiplications are 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 1.54E

The answers of the given multiplication expressed by using scientific notation is $1.932\times {10}^1$.

Explanation of Solution

The given question is,

$\left(9.2\times {10}^{-4}\right)\left(2.1\times {10}^4\right)$

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:

$\begin{array}{r}\left(9.2\times 2.1\right)\left({10}^{-4+4}\right)\\ & =& 19.32\times {10}^0\end{array}$

The above answer is adjusted to a standard position. The decimal is shifted towards one place to the left and the power of $10$ is changed from $0$ to $1$. Hence, the answer of the given multiplication is $1.932\times {10}^1$.

Conclusion

The answer of the given multiplication expressed by using scientific notation is $1.932\times {10}^1$.

Interpretation Introduction

(e)

Interpretation:

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 1.54E

The answer of the given multiplication expressed by using scientific notation is $2.623\times {10}^{12}$.

Explanation of Solution

The given question is,

$\left(4.3\times {10}^6\right)\left(6.1\times {10}^5\right)$

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 the given multiplication can be calculated as follows:

$\begin{array}{r}\left(4.3\times 6.1\right)\left({10}^{6+5}\right)\\ & =& 26.23\times {10}^{11}\end{array}$

The above answer is adjusted to a standard position. The decimal is shifted towards one place to the left and the power of $10$ is changed from $11$ to $12$. Hence, the answer of the given multiplication is $2.623\times {10}^{12}$.

Conclusion

The answer of the given multiplication expressed by using scientific notation is $2.623\times {10}^{12}$.