Class 8 Maths Chapter 10 Exercise 10.2 Exponents and Powers

Understanding Standard Form and Usual Form:

1. Standard form is a way to write very small or large numbers as a product of a number from 1 to 10 and a power of 10.
– Example: 0.0000000000085 becomes 8.5 x 10⁻¹². Move the decimal 12 places to the right.

2. Large numbers have a standard form with a positive exponent.
– Example: 6020000000000000 becomes 6.02 x 10¹⁵. Move the decimal 15 places to the left.

3. To convert from standard form to usual form, move the decimal point.
– Positive exponent (like 10⁶): move decimal to the right.
– Negative exponent (like 10⁻⁶): move decimal to the left.
– Example: 3.02 x 10⁻⁶ becomes 0.00000302.

4. The exponent in standard form indicates how many places the decimal has moved.
– Smaller number: negative exponent.
– Larger number: positive exponent.

5. Standard form is especially useful in science for very precise measurements.

You can apply these steps to express numbers in standard or usual form in their mathematics problems.

Lets have a look at 4 examples that will further clarify the concepts –

Example 1: Express 0.00000000000756 in standard form.
Step 1: Move the decimal 12 places to the right.
Step 2: Result: 7.56 × 10⁻¹².

Example 2: Write 8090000000000000 in standard form.
Step 1: Move the decimal 15 places to the left.
Step 2: Result: 8.09 × 10¹⁵.

Example 3: Convert 4.03 × 10⁻⁴ to usual form.
Step 1: Move the decimal 4 places to the left.
Step 2: Result: 0.000403.

Example 4: Express the size of a bacteria, 0.0000005 m, in standard form.
Step 1: Move the decimal 7 places to the right.
Step 2: Result: 5 × 10⁻⁷ m.

Class 8 Maths Exercise 10.2 Chapter 10 Exponents and Powers

1. Express the following numbers in standard form.

(i) 0.0000000000085 (ii) 0.00000000000942
(iii) 6020000000000000 (iv) 0.00000000837
(v) 31860000000

(i) 0.0000000000085
= 85/10¹³
= (8.5 × 10)/10¹³
= 8.5 × 10 × 10⁻¹³ (1/xᵐ = x⁻ᵐ)
= 8.5 × 10⁽¹⁻¹³⁾ (xᵐ × xⁿ = xᵐ⁺ⁿ)
= 8.5 × 10⁻¹²

(ii) 0.00000000000942
= 942/10¹⁴
= (9.42 × 100)/10¹⁴
= 9.42 × 10² × 10⁻¹⁴ (1/xᵐ = x⁻ᵐ)
= 9.42 × 10²⁻¹⁴ (xᵐ × xⁿ = xᵐ⁺ⁿ)
= 9.42 × 10⁻¹²

(iii) 6020000000000000
= 6.02 × 10¹⁵

(iv) 0.00000000837
= 837/10¹¹
= (8.37 × 100)10¹¹
= 8.37 × 10² × 10⁻¹¹ (1/xᵐ = x⁻ᵐ)
= 8.35 × 10²⁻¹¹ (xᵐ × xⁿ = xᵐ⁺ⁿ)
= 8.37 × 10⁻⁹

(v) 31860000000
= 3.186 × 10¹⁰

2. Express the following numbers in usual form.

(i) 3.02 × 10⁻⁶ (ii) 4.5 × 10⁴ (iii) 3 × 10⁻⁸
(iv) 1.0001 × 10⁹ (v) 5.8 × 10¹² (vi) 3.61492 × 10⁶

(i) 3.02 × 10⁻⁶
= 3.02/10⁶ (x⁻ᵐ = 1/xᵐ)
= 3.02/1000000
= 0.00000302

(ii) 4.5 × 10⁴
= 4.5 × 10000
= 45000

(iii) 3 × 10⁻⁸
= 3/10⁸ (x⁻ᵐ = 1/xᵐ)
= 3/100000000
= 0.00000003

(iv) 1.0001 × 10⁹
= 1.0001 × 1000000000
= 1000100000

(v) 5.8 × 10¹²
= 5.8 × 1000000000000
= 5800000000000

(vi) 3.61492 × 10⁶
= 3.61492 × 1000000
= 3614920

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

(i) 1 micron is equal to 1/1000000 m.
(ii) Charge of an electron is 0.000,000,000,000,000,000,16 coulomb.
(iii) Size of a bacteria is 0.0000005 m
(iv) Size of a plant cell is 0.00001275 m
(v) Thickness of a thick paper is 0.07 mm

(i) 1 micron is equal to 1/1000000 m.
= 1/1000000
= 1/10⁶ = = 1 × 10⁻⁶ m (1/xᵐ = x⁻ᵐ)
1 micron = 1 × 10⁻⁶ m

(ii) Charge of an electron is 0.000,000,000,000,000,000,16 coulomb.
= 16/100000000000000000000
= 16/10²⁰
= (1.6 × 10)/10²⁰
= 1.6 × 10 × 10⁻²⁰
= 1.6 × 10¹⁻²⁰ (xᵐ × xⁿ = xᵐ⁺ⁿ)
Charge of an electron = 1.6 × 10⁻¹⁹ coulomb

(iii) Size of a bacteria is 0.0000005 m
= 5/10000000
= 5/10⁷
= 5 × 10⁻⁷ (1/xᵐ = x⁻ᵐ)
Size of a bacteria = 5 × 10⁻⁷ m

(iv) Size of a plant cell is 0.00001275 m
= 1.275 × 10⁻⁵ m

(v) Thickness of a thick paper is 0.07 mm
= 7/100
= 7 × 10⁻² (1/xᵐ = x⁻ᵐ)
Thickness of a thick paper = 7 × 10⁻² mm

4. In a stack there are 5 books each of thickness 20mm and 5 paper sheets each of thickness 0.016 mm. What is the total thickness of the stack.

Books’ total thickness = 5 × 20 mm = 100 mm
Paper sheets’ total thickness = 5 × 0.016 mm = 0.08 mm
Total thickness = 100 mm + 0.08 mm = 100.08 mm
= 1.0008 × 10² mm
Total thickness of the stack = 1.0008 × 10² mm

Worksheet: Exponents and Powers

Questions

1. Express 0.00000000000756 in standard form.
2. Write 0.0000000000349 in standard form.
3. Convert 8090000000000000 into standard form.
4. Express 0.00000000462 in standard form.
5. Write 5273000000 in standard form.
6. Express 4.03 × 10⁻⁴ in usual form.
7. Convert 7.2 × 10³ into usual form.
8. Write 9 × 10⁻⁷ in usual form.
9. Express 1.0002 × 10⁸ in usual form.
10. Convert 6.5 × 10¹¹ into usual form.
11. Express the diameter of a red blood cell, approximately 0.000007 m, in standard form.
12. Convert the mass of the Earth, about 5972000000000000000000000 kg, into standard form.
13. Write the thickness of a human hair, roughly 0.0001 m, in standard form.
14. Express the distance from Earth to the Sun, around 149600000 km, in standard form.
15. Calculate the total thickness of a stack with 10 books each of 15 mm thickness and 10 sheets of paper each of 0.018 mm thickness.

Answers

1. 7.56 × 10⁻¹²
2. 3.49 × 10⁻¹¹
3. 8.09 × 10¹⁵
4. 4.62 × 10⁻⁹
5. 5.273 × 10⁹
6. 0.000403
7. 7200
8. 0.00000009
9. 100020000
10. 650000000000
11. 7 × 10⁻⁶ m
12. 5.972 × 10²⁴ kg
13. 1 × 10⁻⁴ m
14. 1.496 × 10⁸ km
15. Total thickness = 10 × 15 mm + 10 × 0.018 mm = 150 mm + 0.18 mm = 150.18 mm