MULTIPLYING AND DIVIDING IN SCIENTIFIC NOTATION

Problem 1 :

(4.3 × 10⁸) × (2.0 × 10⁶)

Solution :

= (4.3 × 10⁸) × (2.0 × 10⁶)

To simplify further, we have to make exponents same.

2.0 × 10⁶ = 0.02 × 10² × 10⁶

= 0.02 × 10^{(2 + 6)}

= 0.02 × 10⁸

= (4.3 × 10⁸) × (0.02 × 10⁸)

= (4.3 × 0.02) × 10⁸

= 0.086 × 10⁸

= 8.6 × 10^8-2

= 8.6 × 10⁶

Problem 2 :

(6.0 × 10³) × (1.5 × 10^-2)

Solution :

= (6.0 × 10³) × (1.5 × 10^-2)

= (6 × 10³) × (1.5 × 10^-2)

= (6 × 1.5) × 10^{3 - 2}

= 9 × 10¹

Problem 3 :

 (1.5 × 10^-2) × (8.0 × 10^-1)

Solution :

= (1.5 × 10^-2) × (8.0 × 10^-1)

To simplify further, we have to make exponents same.

8.0 × 10^-1 = 8 × 10^-1 × 10^-1

= 8 × 10^{(-1 - 1)}

= 8 × 10^-2

= (1.5 × 10^-2) × (8 × 10^-2)

= (1.5 × 8) × 10^-2

= 12 × 10^-2

= 1.2 × 10^{-2 + 1}

= 1.2 × 10^-1

Problem 4 :

(7.8 × 10³) / (1.2 × 10⁴)

Solution :

= (7.8 × 10³) / (1.2 × 10⁴)

To simplify further, we have to make exponents same.

7.8 × 10³ = 0.78 × 10¹ × 10³

= 0.78 × 10^{(1 + 3)}

= 0.78 × 10⁴

(7.8 × 10³) / (1.2 × 10⁴) = (0.78 × 10⁴)/(1.2 × 10⁴)

= (0.78/1.2) × 10^{4 + 4}

= 0.65 × 10⁸

= 6.5 × 10^8-1

= 6.5 × 10⁷

Problem 5 :

(8.1 × 10^-2) / (9.0 × 10²)

Solution :

= (8.1 × 10^-2) / (9.0 × 10²)

 (8.1 × 10^-2) / (9.0 × 10²) = (8.1 × 10^-2)/(9 × 10²)

= (8.1/9) × 10^{-2 - 2}

= 0.9 × 10^-4

= 9 × 10^-4-1

= 9 × 10^-5

Problem 6 :

(6.48 × 10⁵) / (2.4 × 10⁴)(1.8 × 10^-2)

Solution :

= (6.48 × 10⁵) / (2.4 × 10⁴)(1.8 × 10^-2)

= (6.48 × 10⁵)/( 4.32 ×10 ²)

= (6.48/4.32) ×10^{5 – 2}

= 1.5 ×10³

Problem 7 :

A gigameter is 1.0 × 10⁶ kilometers. How many square kilometers are in 5 square gigameters?

Solution :

One gigameter = 1.0 × 10⁶ kilometers

Number of square kilometers in 5 gigameter = 5 x 1.0 x 10⁶

= 5.0 x 10⁶

Since we have six at the power, we have to move the decimal six digits to the right.

= 5000000 kilometer

Problem 8 :

The area of the Florida Keys National Marine Sanctuary is about 9.6 × 10³ square kilometers. The area of the Florida Reef Tract is about 16.2% of the area of the sanctuary. What is the area of the Florida Reef Tract in square kilometers?

Solution :

Area of the Florida Key National Marine Santuary = 9.6 × 10³

= 9600 square kilometer

The area of the Florida Reef Tract = 16.2 % of area of sanutuary

= 16.2% of 9600

= 16.2/100 x 9600

= 16.2 x 96

= 1555.2 square kilometer

Area of Florida Reef Tract is approximately 1555 square kilometer.

Problem 9 :

There are about 1.4 × 10⁹ cubic kilometers of water on Earth. About 2.5% of the water is fresh water. How much fresh water is on Earth?

Solution :

Quantity of water in earth = 1.4 × 10⁹ cubic kilometers

Quantity of fresh water = 2.5% of 1.4 × 10⁹

= 2.5/100 x 1.4 × 10⁹

= 2.5 x 1.4 x 10^-2 x 10⁹

= 2.5 x 1.4 x 10^-2+9

= 2.5 x 1.4 x 10⁷

= 3.5 x 10⁷

Moving the decimal seven digits to the right,

= 35000000 cubic kilometers

Problem 10 :

An album has sold 8,780,000 copies. How many more copies does it need to sell to receive the award?

a)  1.22 × 10⁻⁷    b) 1.22 × 10⁻⁶     c) 1.22 × 10⁶     d) 1.22 × 10⁷

Solution :

Use a model to solve the problem. Remaining sales needed for award

= Sales required for award − Current sales total

= 10,000,000 − 8,780,000

= 1,220,000

Converting into scientific notation,

1.22 x 10⁶

The album must sell 1.22 × 10⁶ more copies to receive the award. So, the correct answer is c

Problem 11 :

Use a calculator to evaluate the expression. Write your answer in scientifi c notation and in standard form.

a. (1.5 × 10⁴) + (6.3 × 10⁴)

b. (7.2 × 10⁵) − (2.2 × 10³)

c. (4.1 × 10⁻³) × (4.3 × 10^{− 3})

d. ( 4.75 × 10^{− 6}) × (1.34 × 10⁷)

Solution :

a. (1.5 × 10⁴) + (6.3 × 10⁴)

1.5 × 10⁴ = 15000

6.3 × 10⁴ = 63000

= 15000 + 63000

= 78000

b. (7.2 × 10⁵) − (2.2 × 10³)

7.2 × 10⁵ = 720000

2.2 × 10³ = 2200

= 720000 - 2200

= 717800

c. (4.1 × 10⁻³) × (4.3 × 10^{− 3})

4.1 × 10⁻³ = 0.0041

4.3 × 10⁻³ = 0.0043

= 0.0041(0.0043)

= 0.00001763

d. (4.75 × 10^{− 6}) × (1.34 × 10⁷)

4.75 × 10^{− 6} = 0.00000475

1.34 × 10⁷ = 13400000

= 0.00000475(13400000)

= 63.65