Negative Exponents and Laws of Exponents

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, a^m × aⁿ = a^{m + 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, a^m ÷ aⁿ =a^{m – n}, where m and n are whole numbers and m > n.
• For example,
  2⁵ ÷ 2^{- 6} = 2^{5 - (- 6)} = 2¹¹.

3. Taking Power of a Power:

• For any non-zero integer a, where m and n are whole numbers, (a^m)ⁿ = a^{m × 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, a^m × b^m = (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:

• a^m ÷ b^m = `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).