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;

  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

  1. `2^6`

    Answer: `2^6 = 2 xx 2 xx 2 xx 2 xx 2 xx 2 = 64`
  2. `9^3`

    Answer: `9^3 = 9 xx 9 xx 9 = 729`
  3. `11^2`

    Answer: `11^2 = 11 xx 11 = 121`
  4. `5^4`

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

Question 2: Express the following in exponential form:

  1. `6xx6xx6xx6`

    Answer: `6^4`
  2. `t xx t`

    Answer: `t^2`
  3. `b xx b xx b xx b`

    Answer: `b^4`
  4. `5 xx 5 xx 7 xx 7 xx 7`

    Answer: `5^2 xx 7^3`
  5. `2 xx 2 xx a xx a`

    Answer: `2^2xx\a^2`
  6. `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:

  1. 512

    Answer: `512 = 2 xx 2 xx 2 xx 2 xx 2 xx 2 xx 2 xx 2 xx 2 = 2^9`
  2. 343

    Answer: `343 = 7 xx 7 xx 7 = 7^3`
  3. 729

    Answer: `3 xx 3 xx 3 xx 3 xx 3 xx 3 = 3^6`
  4. 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:

  1. `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`

  2. `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`

  3. `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`

  4. `100^2` or `2^100`

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

  5. `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:

  1. 648

    Answer: `648 = 2 xx 2 xx 2 xx 2 xx 3 xx 3 xx 3`
    `= 2^4 xx 3^3`

  2. 405

    Answer: `405 = 3 xx 3 xx 3 xx 3 xx 5`
    `= 3^4 xx 5`

  3. 540

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

  4. 3600

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

Question 6: Simplify:

  1. `2 xx 10^3`

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

  2. `7^2 xx 2^2`

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

  3. `2^3 xx 5`

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

  4. `3 xx 4^4`

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

  5. `0 xx 10^2`

    Answer: `0 xx 10^2 = 0`

  6. `5^2 xx 3^3`

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

  7. `2^4 xx 3^2`

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

  8. `3^2 xx 10^4`

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

Question 7: Simplify:

  1. `( - 4)^3`

    Answer: `( - 4)^3 = - 64`

  2. `(- 3) xx ( - 2)^3`

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

  3. `( - 3)^2 xx ( - 5)^2`

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

  4. `( - 2)^3 xx ( - 10)^3`

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

Question 8: Compare the following numbers:

  1. `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.

  2. `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.

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