The short notation 104 stands for the product 10×10×10×10. Here ‘10’ is called the base and ‘4’ the exponent. The number 104 is read as 10 raised to the power of 4 or simply as the fourth power of 10. 104 is called the exponential form of 10,000.

The other portion of the chapter gives insight about Laws of exponents. This particular section is divided into several sub-sections:

Multiplying Powers with the Same Base

Dividing Powers with the Same Base

Taking Power of a Power

Multiplying Powers with the Same Exponents

Dividing Powers with the Same Exponents

Numbers with exponent zero

Once the student has come across various laws of exponents miscellaneous examples using the laws of exponents can be studied.

This is followed by Decimal Number System and expressing large numbers in the standard form.

Any number can be expressed as a decimal number between 1.0 and 10.0 including 1.0 MULTIPLIED by a power of 10. Such a form of a number is called its standard form.

The chapter includes 3 unsolved exercises and various solved examples.

The chapter comprises a summary in which all the important concepts of the chapter- Exponents and Powers are mentioned.

Page No 252:

Question 1:

Find the value of:

(i) 26 (ii) 93

(iii) 112 (iv)54

ANSWER:

(i) 26 = 2 × 2 × 2 × 2 × 2 × 2 = 64

(ii) 93 = 9 × 9 × 9 = 729

(iii) 112 = 11 × 11 = 121

(iv)54 = 5 × 5 × 5 × 5 = 625

Page No 252:

Question 2:

Express the following in exponential form:

(i) 6 × 6 × 6 × 6 (ii) t × t

(iii) b × b × b × b (iv) 5 × 5 × 7 ×7 × 7

(v) 2 × 2 × a × a (vi) a × a × a × c × c × c × c × d

ANSWER:

(i) 6 × 6 × 6 × 6 = 64

(ii) t × t= t2

(iii) b × b × b × b = b4

(iv) 5 × 5 × 7 × 7 × 7 = 52 × 73

(v) 2 × 2 × a × a = 22 × a2

(vi) a × a × a × c × c × c × c × d = a3 c4 d

Page No 253:

Question 3:

Express the following numbers using exponential notation:

(i) 512 (ii) 343

(iii) 729 (iv) 3125

ANSWER:

(i) 512 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 29

(ii) 343 = 7 × 7 × 7 = 73

(iii) 729 = 3 × 3 × 3 × 3 × 3 × 3 = 36

(iv) 3125 = 5 × 5 × 5 × 5 × 5 = 55

Page No 253:

Question 4:

Identify the greater number, wherever possible, in each of the following?

(i) 43 or 34 (ii) 53 or 35

(iii) 28 or 82 (iv) 1002 or 2100

(v) 210 or 102

ANSWER:

(i) 43 = 4 × 4 × 4 = 64

34 = 3 × 3 × 3 × 3 = 81

Therefore, 34 > 43

(ii) 53 = 5 × 5 × 5 =125

35 = 3 × 3 × 3 × 3 × 3 = 243

Therefore, 35 > 53

(iii) 28 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 256

82 = 8 × 8 = 64

Therefore, 28 > 82

(iv)1002 or 2100

210 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 1024

2100 = 1024 × 1024 × 1024 × 1024 × 1024 × 1024 × 1024 × 1024 ×1024 × 1024

1002 = 100 × 100 = 10000

Therefore, 2100 > 1002

(v) 210 and 102

210 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 1024

102 = 10 × 10 = 100

Therefore, 210 > 102

Page No 253:

Question 5:

Express each of the following as product of powers of their prime factors:

(i) 648 (ii) 405

(iii) 540 (iv) 3,600

ANSWER:

(i) 648 = 2 × 2 × 2 × 3 × 3 × 3 × 3 = 23. 34

(ii) 405 = 3 × 3 × 3 × 3 × 5 = 34 . 5

(iii) 540 = 2 × 2 × 3 × 3 × 3 × 5 = 22. 33. 5

(iv) 3600 = 2 × 2 × 2 × 2 × 3 × 3 × 5 × 5 = 24. 32. 52

Page No 253:

Question 6:

Simplify:

(i) 2 × 103 (ii) 72 × 22

(iii) 23 × 5 (iv) 3 × 44

(v) 0 × 102 ­­­­­ (vi) 52 × 33

(vii) 24 × 32 (viii) 32 × 104

ANSWER:

(i) 2 × 103 = 2 × 10 × 10 × 10 = 2 × 1000 = 2000

(ii) 72 × 22 = 7 × 7 × 2 × 2 = 49 × 4 = 196

(iii) 23 × 5 = 2 × 2 × 2 × 5 = 8 × 5 = 40

(iv) 3 × 44 = 3 × 4 × 4 × 4 × 4 = 3 × 256 = 768

(v) 0 × 102 = 0 × 10 × 10 = 0

(vi) 52 × 33 = 5 × 5 × 3 × 3 × 3 = 25 × 27 = 675

(vii) 24 × 32 = 2 × 2 × 2 × 2 × 3 × 3 = 16 × 9 = 144

(viii) 32 × 104 = 3 × 3 × 10 × 10 × 10 × 10 = 9 × 10000 = 90000

Page No 253:

Question 7:

Simplify:

(i) (− 4)3 (ii) (− 3) × (− 2)3

(iii) (− 3)2 × (− 5)2 (iv)(− 2)3 × (−10)3

ANSWER:

(i) (−4)3 = (−4) × (−4) × (−4) = −64

(ii) (−3) × (−2)3 = (−3) × (−2) × (−2) × (−2) = 24

(iii) (−3)2 × (−5)2 = (−3) × (−3) × (−5) × (−5) = 9 × 25 = 225

(iv) (−2)3 × (−10)3 = (−2) × (−2) × (−2) × (−10) × (−10) × (−10)

= (−8) × (−1000) = 8000

Page No 253:

Question 8:

Compare the following numbers:

(i) 2.7 × 1012; 1.5 × 108

(ii) 4 × 1014; 3 × 1017

ANSWER:

(i) 2.7 × 1012; 1.5 × 108

2.7 × 1012 > 1.5 × 108

(ii) 4 × 1014; 3 × 1017

3 × 1017 > 4 × 1014

Page No 260:

Question 1:

Using laws of exponents, simplify and write the answer in exponential form:

(i) 32 × 34 × 38 (ii) 615 ÷ 610 (iii) a3 × a2

(iv) 7X× 72 (v) (vi) 25 × 55

(vii) a4 × b4 (viii) (34)3

(ix) (x) 8t ÷ 82

ANSWER:

(i) 32 × 34 × 38 = (3)2 + 4 + 8 (am × an = am+n)

= 314

(ii) 615 ÷ 610 = (6)15 − 10 (am ÷ an = am−n)

= 65

(iii) a3 × a2 = a(3 + 2) (am × an = am+n)

= a5

(iv) 7x + 72 = 7x + 2 (am × an = am+n)

(v) (52)3 ÷ 53

= 52 × 3 ÷ 53 (am)n = amn

= 56 ÷ 53

= 5(6 − 3) (am ÷ an = am−n)

= 53

(vi) 25 × 55

= (2 × 5)5 [am × bm = (a × b)m]

= 105

(vii) a4 × b4

= (ab)4 [am × bm = (a × b)m]

(viii) (34)3 = 34 × 3 = 312 (am)n = amn

(ix) (220 ÷ 215) × 23

= (220 − 15) × 23 (am ÷ an = am−n)

= 25 × 23

= (25 + 3) (am × an = am+n)

= 28

(x) 8t ÷ 82 = 8(t − 2) (am ÷ an = am−n)