# Negative Exponents and Laws of Exponents

## Notes

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