Chapter 3, Problem 14CR

(a)

Interpretation Introduction

Interpretation : To convert the given number 0.0008917 to standard scientific notation.

Concept Introduction : Scientific notation for a number is written in such a way that large numbers are written in small decimal form which is then multiplied by the power of 10.

For numbers less than 1, the power of 10 in scientific notation has negative exponent and for numbers greater than 1, the power is either zero or has positive exponent.

Answer to Problem 14CR

  $8.917\times {10}^{-4}$

Explanation of Solution

In order to convert numbers to scientific notation, we must follow the following rules:

Use a single number before the decimal point. All other numbers will be placed after the decimal point.

The basis of expression will be the 10^th power. The powers can be any integer number.

The given number: $0.0008917$

Must be converted, so select the first number, ensure the power is correct.

  $8.917$ will be the basis. Since, the decimal shifts 4 positions to the right, the power of ten will be $-4$ . Thus, the scientific notation is $8.917\times {10}^{-4}$ .

(b)

Interpretation Introduction

Interpretation : To convert the given number $2.795\times {10}^{-4}$ to ordinary decimal notation.

Concept Introduction : Scientific notation for a number is written in such a way that large numbers are written in small decimal form which is then multiplied by the power of 10.

For numbers less than 1, the power of 10 in scientific notation has negative exponent and for numbers greater than 1, the power is either zero or has positive exponent.

Answer to Problem 14CR

  $0.0002795$

Explanation of Solution

In order to convert expressions in scientific notation to ordinary decimal notation, we must follow the following rules:

Multiply the given power to the base number.

  $2.7595\times {10}^{-4}=0.0002795$

When multiplied, we get a decimal number, typically larger than the original expression.

(c)

Interpretation Introduction

Interpretation : To convert the given number $4.913\times {10}^3$ to ordinary decimal notation.

Concept Introduction : Scientific notation for a number is written in such a way that large numbers are written in small decimal form which is then multiplied by the power of 10.

For numbers less than 1, the power of 10 in scientific notation has negative exponent and for numbers greater than 1, the power is either zero or has positive exponent.

Answer to Problem 14CR

  $4913$

Explanation of Solution

In order to convert expressions in scientific notation to ordinary decimal notation, we must follow the following rules:

Multiply the given power to the base number.

  $4.913\times {10}^3=4913$

When multiplied, we get a decimal number, typically larger than the original expression.

(d)

Interpretation Introduction

Interpretation : To convert the given number 85,100,000 to standard scientific notation.

Concept Introduction : Scientific notation for a number is written in such a way that large numbers are written in small decimal form which is then multiplied by the power of 10.

For numbers less than 1, the power of 10 in scientific notation has negative exponent and for numbers greater than 1, the power is either zero or has positive exponent.

Answer to Problem 14CR

  $8.51\times {10}^7$

Explanation of Solution

In order to convert numbers to scientific notation, we must follow the following rules:

Use a single number before the decimal point. All other numbers will be placed after the decimal point.

The basis of expression will be the 10^th power. The powers can be any integer number.

The given number:

  $85100000$

Must be converted, so select the first number, ensure the power is correct.

  $8.51$ will be the basis. Since, the decimal moves 7 steps to the left, the power of tenth will be+7.

Thus, the scientific notation is $8.51\times {10}^7$ .

(e)

Interpretation Introduction

Interpretation : To convert the given calculation $5.751\times {10}^5\times 2.119\times {10}^{-4}$ to standard scientific notation.

Concept Introduction : Scientific notation for a number is written in such a way that large numbers are written in small decimal form which is then multiplied by the power of 10.

For numbers less than 1, the power of 10 in scientific notation has negative exponent and for numbers greater than 1, the power is either zero or has positive exponent.

Answer to Problem 14CR

  $1.219\times {10}^2$

Explanation of Solution

In order to convert numbers to scientific notation, we must follow the following rules:

Use a single number before the decimal point. All other numbers will be placed after the decimal point.

The basis of expression will be the 10^th power. The powers can be any integer number.

For a multiplication or division, significant figures rules must be followed.

The total amount of significant figures in the final result must be equal to the least amount of significant figures in the numbers multiplied or divided.

From the first number, 4 significant figures

From the second number, 4 significant figures.

Then, 4 significant figures must be kept.

The given number:

Calculate the basis numbers $\left(5.751\times 2.119\right)=12.186369$

Multiply the powers:

The tenth base:

  ${10}^{\left(5-4\right)}={10}^1=10$

Thus, the number becomes 121.863

Keep only 4 significant numbers 121.863, bold digitmust be rounded up. Since, it is followed by digit 6 which is greater than 5, the digit 8 is rounded up to 9. The number becomes 121.9.

Thus, the scientific figure will be $1.219\times {10}^2$

(f)

Interpretation Introduction

Interpretation : To convert the given calculation to standard scientific notation.

Concept Introduction : Scientific notation for a number is written in such a way that large numbers are written in small decimal form which is then multiplied by the power of 10.

For numbers less than 1, the power of 10 in scientific notation has negative exponent and for numbers greater than 1, the power is either zero or has positive exponent.

Answer to Problem 14CR

  $3.396\times {10}^{-9}$

Explanation of Solution

In order to convert numbers to scientific notation, we must follow the following rules:

Use a single number before the decimal point. All other numbers will be placed after the decimal point.

The basis of expression will be the 10^th power. The powers can be any integer number.

For a multiplication or division, significant figures rules must be followed.

The total amount of significant figures in the final result must be equal to the least amount of significant figures in the numbers multiplied or divided.

From the first number, 4 significant figures.

From the second number, 4 significant figures.

Then, significant figures must be kept.

The given number:

Calculate the basis numbers $\left(\frac{2.791}{8.219}\right)=0.3395790$

Dividing the powers:

The tenth base:

  ${10}^{\left(-5-3\right)}={10}^{-8}$

Keep only 4 significant numbers $0.3395790$ , 5 must be rounded up to $0.3396$

  $0.3396=3.396\times {10}^{-1}$

Final expression

  $=\left(3.396\times {10}^{-1}\right)\left({10}^{-8}\right)=3.396\times {10}^{-9}$

Thus, the scientific notation will be $3.396\times {10}^{-9}$ .