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 = 10^4`
(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;

• `a^m × a^n = a^(m + n)`
• `a^m ÷ a^n = a^(m - n)`
• `(a^m)^n = a^(mn)`
• `a^m × b^m = (ab)^m`
• 
• 
• (- 1)^{even number} = 1 and (- 1)^{odd number} = - 1

Exercise 13.1

Question 1: Find the value of

• `2^6`
  
  Answer: `2^6 = 2 xx 2 xx 2 xx 2 xx 2 xx 2 = 64`
• `9^3`
  
  Answer: `9^3 = 9 xx 9 xx 9 = 729`
• `11^2`
  
  Answer: `11^2 = 11 xx 11 = 121`
• `5^4`
  
  Answer: `5^4 = 5 xx 5 xx 5 xx 5 = 625`

Question 2: Express the following in exponential form:

• `6xx6xx6xx6`
  
  Answer: `6^4`
• `t xx t`
  
  Answer: `t^2`
• `b xx b xx b xx b`
  
  Answer: `b^4`
• `5 xx 5 xx 7 xx 7 xx 7`
  
  Answer: `5^2 xx 7^3`
• `2 xx 2 xx a xx a`
  
  Answer: `2^2xx\a^2`
• `a xx a xx a xx c xx c xx c xx c xx d`
  
  Answer: `a^3xx\c^4xx\d^1`

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

• 512
  
  Answer: `512 = 2 xx 2 xx 2 xx 2 xx 2 xx 2 xx 2 xx 2 xx 2 = 2^9`
• 343
  
  Answer: `343 = 7 xx 7 xx 7 = 7^3`
• 729
  
  Answer: `3 xx 3 xx 3 xx 3 xx 3 xx 3 = 3^6`
• 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:

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

• 648
  
  Answer: `648 = 2 xx 2 xx 2 xx 2 xx 3 xx 3 xx 3`
  `= 2^4 xx 3^3`
  
  
• 405
  
  Answer: `405 = 3 xx 3 xx 3 xx 3 xx 5`
  `= 3^4 xx 5`
  
  
• 540
  
  Answer: `540 = 2^2 xx 3^3 xx 5`
  
  
• 3600
  
  Answer: `3600 = 2^4 xx 3^2 xx 5^2`

Question 6: Simplify:

• `2 xx 10^3`
  
  Answer: `2 xx 10^3 = 2 xx 1000 = 2000`
  
  
• `7^2 xx 2^2`
  
  Answer: `7^2 xx 2^2 = 49 xx 4 = 196`
  
  
• `2^3 xx 5`
  
  Answer: `2^3 xx 5 = 8 xx 5 = 40`
  
  
• `3 xx 4^4`
  
  Answer: `3 xx 4^4 = 3 xx 256 = 768`
  
  
• `0 xx 10^2`
  
  Answer: `0 xx 10^2 = 0`
  
  
• `5^2 xx 3^3`
  
  Answer: `5^2 xx 3^3 = 25 xx 27 = 675`
  
  
• `2^4 xx 3^2`
  
  Answer: `2^4 xx 3^2 = 16 xx 9 = 144`
  
  
• `3^2 xx 10^4`
  
  Answer: `3^2 xx 10^4 = 9 xx 10000 = 90000`

Question 7: Simplify:

• `( - 4)^3`
  
  Answer: `( - 4)^3 = - 64`
  
  
• `(- 3) xx ( - 2)^3`
  
  Answer: `( - 3) xx ( - 2)^3 = ( - 3) xx ( - 8) = 24`
  
  
• `( - 3)^2 xx ( - 5)^2`
  
  Answer: `( - 3)^2 xx ( - 5)^2 = 9 xx 25 = 225`
  
  
• `( - 2)^3 xx ( - 10)^3`
  
  Answer: `( - 2)^3 xx ( - 10)^3 = ( - 8)xx ( - 1000) = 8000`

Question 8: Compare the following numbers:

• `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.
  
  
• `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.