ML Aggarwal: Exponents & Powers - 2

Q.1. Express the following numbers in standard form:
(i) 0.0000000000085
(ii) 0.000000000000942
(iii) 6020000000000000
(iv) 0.00000000837

Let us express the numbers in their standard form,
(i) 0.0000000000085

0.0000000000085 = 8.5 × 10^-12

(ii) 0.000000000000942

0.000000000000942 = 9.42 × 10^-13

(iii) 6020000000000000

6020000000000000 = 6.02 × 10¹⁵

(iv) 0.00000000837

0.00000000837 = 8.37 × 10^-9

Q.2. Express the following numbers in usual form:
(i) 3.02 × 10^-6
(ii) 1-007 × 10¹¹
(iii) 5.375 × 10¹⁴
(iv) 7.579 × 10-¹⁴

Let us express the numbers in their usual form,
(i) 3.02 × 10^-6

3.02 × 10^-6 = 0.00000302

(ii) 1-007 × 10¹¹

1.007 × 10¹¹ = 100700000000

(iii) 5.375 × 10¹⁴

5.375 × 10¹⁴ = 537500000000000 

(iv) 7.579 × 10-¹⁴

7.579 × 10-¹⁴ = 0.00000000000007579

Q.3. Express the number appearing in the following statements in standard form:

(i) The mass of a proton is 0.000000000000000000000001673 gram.

 The mass of a proton is 0.000000000000000000000001673 gram, it is expressed in standard form as 1.673 × 10-²⁴ gram.

(ii) The thickness of a piece of paper is 0.0016 cm.

Thickness of a piece of paper in standard form is 0.0016 cm; it is expressed in standard form as 1.6 × 10^-3

(iii) The diameter of a wire on a computer chip is 0.000003 m.

Diameter of a wire on a computer chip is 0.000003 m; it is expressed in standard form as 3.0 × 10^-6 m

(iv) A helium atom has a diameter of m.

A helium atom has a diameter of 22/100000000000 m; it is expressed in standard form as
22 × 10^-12 = 2.2 × 10^-10

(v) Mass of a molecule of hydrogen gas is about 0.00000000000000000000334 tons.

Mass of a molecule of hydrogen gas is about 0.00000000000000000000334 tons; it is expressed in standard form as = 3.34 × 10-²¹ tons

(vi) The human body has 1 trillion cells which vary in shapes and sizes.

Human body has 1 trillion of cells which vary in shapes and sizes; it is expressed in standard form as 1,000,000,000,000 = 10¹²

(vii) The distance from the Earth of the Sun is 149,600,000,000 m.

The distance from the Earth of the Sun is expressed in standard form as
149,600,000,000 m = 1.496 × 10¹¹

(viii) The speed of light is 300,000,000 m/sec

The speed of light is 300,000,000 m/sec; it is expressed in standard form as 3.0 × 10⁸ m/sec

(ix) Mass of the Earth is 5,970,000,000,000,000,000,000,000 kg.

Mass of the Earth is 5,970,000,000,000,000,000,000,000 kg; it is expressed in standard form as 5.97 × 10²⁴ kg

(x) Express 3 years in seconds.

Express 3 years in seconds, it is expressed in standard form as 3 years = 3 × 365 days
= 3 × 365 × 24 hours
= 3 × 365 × 24 × 3600 seconds
= 1040688000 seconds
= 1.040688 × 10⁹ seconds

(xi) Express 7 hectares in cm².

Express 7 hectares in cm², it is expressed in standard form as 7 hectares = 7 × 10000 m²
= 7 × 10000 × 100 × 100 cm²
= 700000000 cm2 = 7.0 × 108 cm²

(xii) A sugar factory has annual sales of 3 billion 720 million kilograms of sugar.

A sugar factory has annual sales of 3 billion 720 million kilograms of sugar, it is expressed in standard form as
Annual sale of a sugar factory = 3 billion
720 million kilograms sugar = 3,720,000,000 kg = 3.72 × 109 kg

Q.4. Compare the following:
(i) Size of a plant cell to the thickness of a piece of paper.
(ii) Size of a plant cell to the diameter of a wire on a computer chip.
(iii) The thickness of a piece of paper to the diameter of a wire on a computer chip.
Given size of plant cell = 0.00001275 m
Thickness of a piece of paper = 0.0016 cm
Diameter of a wire on a computer chip = 0.000003 m

Given:
Size of plant cell= 0.00001275 m = 1.275 × 10^-5 m
Thickness of a piece of paper = 0.0016 cm = 1.6 × 10^-3 cm
Diameter of a wire on a computer chip = 0.000003 m = 3.0 × 10^-6 m

(i) Size of plant cell: thickness of a piece of paper

1.275 × 10-5: 1.6 × 10^-3
Size of plant cell = 1.2/1.6 = 3/4 times of thickness of paper.

(ii) Size of plant cell: diameter of wire on a computer chip 

1.275 × 10^-5: 3.0 × 10^-6
12.75: 3.00
Size of plant cell is 4 times of diameter of wire.

(iii) Thickness of a piece of paper: diameter of a wire on a computer chip 

1.6 × 10-3: 3.0 × 10-6 × 100 cm
1.6 × 1000: 300
16.1: 3
Approximately 5 times is the thickness of paper to diameter of wire.

Q.5. The number of red blood cells per cubic millimeter of blood is approximately 5.5 million. If the average body contains 5 liters of blood, what is the total number of red cell in the body? (1 liter = 1,00,000 mm³)

Given:
Red blood per cubic millimeter = 5.5 million = 5.5 × 10⁶
Red blood in 5 liters of blood = 5.5 × 10⁶ × 5 × 10⁵ (1 litre = 10⁵ mm)
= 27.5 × 10⁶+5
= 27.5 × 10¹¹
= 2.75 × 10 × 10¹¹
= 2.75 × 10¹²
∴Total number of red blood cells in the body is 2.75 × 10¹². 

Q.6. Mass of Mars is 6.42 × 10²⁹ kg and the mass of the sun is 1.99 × 10³⁰ kg. What is the total mass?

Given:
Mass of Mars = 6.42 × 10²⁹ kg
and mass of sun = 1.99 × 10³⁰
Total mass = 6.42 × 10²⁹ + 1.99 × 10³⁰
= 10²⁹ (6.42 + 1.99 × 10)
= 10²⁹ (6.42 + 19.9)
= 26.32 × 10²⁹
∴ Total mass of mars is 26.32 × 1029.

Q.7. A particular star is at a distance of about 8.1 × 10¹³ km from the Earth. Assuming that the light travels at 3 × 10⁸ m/sec, find how long does light take from that star to reach the Earth.

Given:
Distance between earth and a particular star = 8.1 × 10¹³ km
Speed of light = 3 × 108 m/sec.
Time is taken to reach the earth =
= 2.7 × 10^16-8
= 2.7 × 10⁸ sec
∴  Light takes 2.7 × 10⁸ sec from star to reach the earth.