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 รท 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.