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Negative Exponents and Laws of Exponents

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Negative Exponents and Laws of Exponents:

1. Multiplying Powers With the Same Base:

  • For any non-zero integer a, where m and n are whole numbers, am × an = am + n.
  • This law is also used when the exponents are negative.
  • For example,
    `(-3)^(-4) xx (-3)^(-3)`
    ` = 1/(-3)^4 xx 1/(-3)^3`
    `= 1/((-3)^4 xx (-3)^3)`
    `= 1/((-3)^(4 + 3))`
    `= (-3)^(-7)`

2. Dividing Powers with the Same Base:

  • In general, for any non-zero integer a, am ÷ an =am – n, where m and n are whole numbers and m > n.
  • For example,
    25 ÷ 2- 6 = 25 - (- 6) = 211.

3. Taking Power of a Power:

  • For any non-zero integer a, where m and n are whole numbers, (am)n = am × n.
  • For example,
    `{(-2/3)^-2}^2 = (-2/3)^(-2 xx 2) = (-2/3)^(-4)`

4. Multiplying Powers with the Same Exponents:

  • In general, for any non-zero integer a, am × bm = (ab)m where m is any whole number.
  • Example,
    2- 4 × (-3)- 4 = `(2 xx (-3))^(- 4) = - 6^(-4).`

5. Dividing Powers with the Same Exponents:

  • am ÷ bm = `a^m/b^m = (a/b)^m`, where a, and b are any non-zero integers and m is a whole number.
  • For example,
    6- 3 ÷ 5- 3 = `(6^-3)/(5^-3) = (6/5)^-3`.

6. Numbers with exponent zero:

  • Any number (except 0) raised to the power (or exponent) 0 is 1.
  • For example,
    7-3 ÷ 7-3 = 7(-3 - (- 3)) = 7(-3 + 3) = 70 = 1.
    Thus, a0 = 1.....(for any non-zero integer a).
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Shaalaa.com | Problems on Exponents & Powers - Part 1

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Problems on Exponents & Powers - Part 1 [00:12:07]
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