Class 7 Maths

# Exponents and Powers

## Introduction

It is difficult to read, understand, compare and operate with very large numbers. Exponents are used for expressing very large numbers in shorter forms.

Example: 10,000 = 104
(It is read as 10 raised to 4)
Here, 10 is called the base and 4 is the exponent.

### Laws of Exponents

For any non-zero integers a and b and whole numbers m and n:

1. a^m × a^n = a^(m + n)
2. a^m รท a^n = a^(m - n)
3. (a^m)^n = a^(mn)
4. a^m × b^m = (ab)^m
5. a^m÷b^m=(a/b)^m
6. a^0=1
7. (- 1)even number = 1 and (- 1)odd number = - 1

## Exercise 13.1

Question 1: Find the value of

(a) 2^6

Answer: 2^6 = 2 xx 2 xx 2 xx 2 xx 2 xx 2 = 64

(b) 9^3

Answer: 9^3 = 9 xx 9 xx 9 = 729

(c) 11^2

Answer: 11^2 = 11 xx 11 = 121

(d) 5^4

Answer: 5^4 = 5 xx 5 xx 5 xx 5 = 625

Question 2: Express the following in exponential form:

1. 6xx6xx6xx6
2. t xx t
3. b xx b xx b xx b
4. 5 xx 5 xx 7 xx 7 xx 7
5. 2 xx 2 xx a xx a
6. a xx a xx a xx c xx c xx c xx c xx d

Answer: (a), 6^4 (b), t^2 (c), b^4 (d), 5^2 xx 7^3 (e), 2^2xx\a^2 (f) a^3xx\c^4xx\d^1

Question 3: Express each of the following numbers using exponential notation:

(a) 512

Answer: 512 = 2 xx 2 xx 2 xx 2 xx 2 xx 2 xx 2 xx 2 xx 2 = 2^9

(b) 343

Answer: 343 = 7 xx 7 xx 7 = 7^3

(c) 729

Answer: 3 xx 3 xx 3 xx 3 xx 3 xx 3 = 3^6

(d) 3125

Answer: 3125 = 5 xx 5 xx 5 xx 5 xx 5 = 5^5

Question 4: Identify the greater number, wherever possible, in each of the following:

(a) 4^3 or 3^4

Answer: 4^3 = 4 xx 4 xx 4 = 64

3^4 = 3 xx 3 xx 3 xx 3 = 81

Hence, 4^3 < 3^4

(b) 5^3 or 3^5

Answer: 5^3 = 5 xx 5 xx 5 = 125

3^5 = 3 xx 3 xx 3 xx 3 xx 3 = 243

Hence, 5^3 < 3^5

(c) 2^8 or 8^2

Answer: 2^8 = 2 xx 2 xx 2 xx 2 xx 2 xx 2 xx 2 xx 2 = 256

8^2 = 64

Hence, 2^8 > 8^2

(d) 100^2 or 2^100

Answer: 2^100 > 100^2 (because exponent is much larger for base 2.)

(e) 2^10 or 10^2

Answer: 2^10 > 10^2 (because exponent is larger for base 2.)

Question 5: Express each of the following as product of powers of their prime factors:

(a) 648

Answer: 648 = 2 xx 2 xx 2 xx 2 xx 3 xx 3 xx 3

= 2^4 xx 3^3

(b) 405

Answer: 405 = 3 xx 3 xx 3 xx 3 xx 5

= 3^4 xx 5

(c) 540

Answer: 540 = 2^2 xx 3^3 xx 5

(d) 3600

Answer: 3600 = 2^4 xx 3^2 xx 5^2

Question 6: Simplify:

(a) 2 xx 10^3

Answer: 2 xx 10^3 = 2 xx 1000 = 2000

(b) 7^2 xx 2^2

Answer: 7^2 xx 2^2 = 49 xx 4 = 196

(c) 2^3 xx 5

Answer: 2^3 xx 5 = 8 xx 5 = 40

(d) 3 xx 4^4

Answer: 3 xx 4^4 = 3 xx 256 = 768

(e) 0 xx 10^2

Answer: 0 xx 10^2 = 0

(f) 5^2 xx 3^3

Answer: 5^2 xx 3^3 = 25 xx 27 = 675

(g) 2^4 xx 3^2

Answer: 2^4 xx 3^2 = 16 xx 9 = 144

(h) 3^2 xx 10^4

Answer: 3^2 xx 10^4 = 9 xx 10000 = 90000

Question 7: Simplify:

(a) ( - 4)^3

Answer: ( - 4)^3 = - 64

(b) (- 3) xx ( - 2)^3

Answer: ( - 3) xx ( - 2)^3 = ( - 3) xx ( - 8) = 24

(c) ( - 3)^2 xx ( - 5)^2

Answer: ( - 3)^2 xx ( - 5)^2 = 9 xx 25 = 225

(d) ( - 2)^3 xx ( - 10)^3

Answer: ( - 2)^3 xx ( - 10)^3 = ( - 8)xx ( - 1000) = 8000

Question 8: Compare the following numbers:

(a) 2.7 xx 10^12 and 1.5 xx 10^8

Answer: 1.5 xx 10^8 < 2.7 xx 10^12

Because exponent on 10 is larger in case of first number.

(b) 4 xx 10^14 and 3 xx 10^17

Answer: 4 xx 10^14 < 3 xx 10^17

Because exponent on 10 is smaller in case of first number.