ADDITION AND SUBTRACTION IN SCIENTIFIC NOTATION

Problem 1 :

(1.2 × 10⁵) + (5.35 × 10⁶)

Solution :

= (1.2 × 10⁵) + (5.35 × 10⁶)

In order to simplify further, we have to make exponents same.

1.2 × 10⁵ = 0.12 × 10¹ × 10⁵

By combining the powers using a^m ⋅ aⁿ = a^{m + n} , we get

= 0.12 × 10^{(1+ 5)}

= 0.12 × 10⁶

1.2 × 10⁵ = 0.12 × 10⁶  ------(1)

5.35 × 10⁶ = 5.35 × 10⁶   ------(2)

(1) + (2)

= (0.12 × 10⁶) + (5.35 × 10⁶)

= (0.12 + 5.35) × 10⁶

= 5.47 × 10⁶

Problem 2 :

(6.91 × 10^-2) + (2.4 × 10^-3)

Solution :

= (6.91 × 10^-2) + (2.4 × 10^-3)

To simplify further, we have to make exponents same.

6.91 × 10^-2 = 69.1 ×10^-1 ×10^-2

= 69.1 ×10^{(-1 – 2)}

= 69.1 ×10^-3

6.91 × 10^-2 = 69.1 ×10^-3 ------(1)

2.4 × 10^-3 = 2.4 × 10^-3 ------(2)

(1) + (2)

= (69.1 ×10^-3) + (2.4 ×10^-3)

= (69.1+ 2.4) ×10^-3

= 71.5 ×10^-3

= 7.15 ×10^{-3 + 1}

= 7.15 ×10^-2

Problem 3 :

(9.70 × 10⁶) + (8.3 × 10⁵)

Solution :

= (9.70 × 10⁶) + (8.3 × 10⁵)

To simplify further, we have to make exponents same.

8.3 × 10⁵ = 0.83 × 10¹ × 10⁵

By combining the powers using a^m ⋅ aⁿ = a^{m + n} , we get

= 0.83 × 10^{(1+ 5)}

= 0.83 × 10⁶

9.70 × 10⁶ = 9.70× 10⁶ ------(1)

8.3 ×10⁵ = 0.83 × 10⁶ ------(2)

(1) + (2)

= (9.70 × 10⁶) + (0.83 × 10⁶)

= (9.70 + 0.83) × 10⁶

= 10.53 × 10⁶

= 1.053 × 10⁷

Problem 4 :

(3.67 × 10²) - (1.6 × 10¹)

Solution :

(3.67 × 10²) - (1.6 × 10¹)

In order to simplify further, we have to make exponents same.

1.6 × 10¹ = 0.16 × 10¹ × 10¹

By combining the powers using a^m ⋅ aⁿ = a^{m + n} , we get

= 0.16 × 10^{(1+ 1)}

= 0.16 × 10² 

3.67 × 10² = 3.67 × 10² ------(1)

1.6 × 10¹ = 0.16 × 10²  ------(2)

(1) - (2)

= (3.67 × 10²) - (0.16 × 10²)

= (3.67 - 0.16) × 10²

= 3.51 × 10²

Problem 5 :

(8.41 × 10^-5) - (7.9 × 10^-6)

Solution :

(8.41 × 10^-5) - (7.9 × 10^-6)

In order to simplify further, we have to make exponents same.

8.41 × 10^-5 = 84.1 ×10^-1 ×10^-5

By combining the powers using a^m ⋅ aⁿ = a^{m + n} , we get

= 84.1 ×10^{(-1 – 5)}

= 84.1 ×10^-6

8.41 × 10^-5 = = 84.1 ×10^-6  ------(1)

 7.9 × 10^-6 = 7.9 × 10^-6 ------(2)

(1) - (2)

= (84.1 ×10^-6) - (7.9 ×10^-6)

= (84.1- 7.9) ×10^-6

= 76.2 ×10^-6

= 7.62 ×10^-6+1

= 7.62 ×10^-5

Problem 6 :

(1.33 × 10⁵) - (4.9 × 10⁴)

Solution :

= (1.33 × 10⁵) - (4.9 × 10⁴)

In order to simplify further, we have to make exponents same.

4.9 × 10⁴ = 0.49 × 10¹ × 10⁴

By combining the powers using a^m  ⋅  aⁿ = a^{m + n} , we get

= 0.49 × 10^{(1+ 4)}

= 0.49 × 10⁵ 

1.33 × 10⁵ = 1.33 × 10⁵  ------(1)

4.9 × 10⁴ = 0.49 × 10⁵ ------(2)

(1) - (2)

= (1.33 × 10⁵) - (0.49 × 10⁵)

= (1.33 - 0.49) × 10⁵

= 0.84 × 10⁵

= 8.4 × 10^5-1

= 8.4 × 10⁴

Problem 7 :

According to scientists, the Earth's mass is 5.98 x 10²⁴ kilograms. The mass of the Sun is 1.989 x 10³⁰ kilograms. How much greater is the mass of the sun than the mass of the Earth ?

Solution :

Earth's mass = 5.98 x 10²⁴ kilograms

Mass of the Sun = 1.989 x 10³⁰ kilograms

Difference = 1.989 x 10³⁰ - 5.98 x 10²⁴

= 1.989 x 10⁶ x 10²⁴ - 5.98 x 10²⁴

= 1989000 x 10²⁴ - 5.98 x 10²⁴

= (1989000 - 5.98) x 10²⁴

= 1988994.02 x 10²⁴

= 1.98899402 x 10^-6 x 10²⁴ 

= 1.98899402 x 10^-6+24

= 1.98899402 x 10¹⁸

Problem 8 :

Last year, we noticed that the population of Tam worth was 5.6 x 10³. The population Liverpool was 1.3 x 10⁴. Which town had a larger population and by how much ?

Solution :

Population of Tam worth = 5.6 x 10³

Population of Liverpool = 1.3 x 10⁴

Converting into standard form,

Population of Tam worth = 5600

Population of Liverpool = 13000

Difference = 13000 - 5600

= 7400

Liver pool has larger population and it is larger than 7400.

Problem 9 :

How many times bigger is the distance from Earth to the sun of 9.3 x 10⁶ miles than the furthest distance from Earth to the moon of 3 x 10⁶ miles.

Solution :

Distance from Earth to the sun of 9.3 x 10⁶ miles

Furthest distance from Earth to the moon of 3 x 10⁶ miles

Number of times =  9.3 x 10⁶ /  3 x 10⁶

= 3.1 x 10^{6 - 6}

= 3.1 x 10⁰

= 3.1 x 1

= 3.1 times