Exponents

Lesson on Exponents and Their Rules

Exponents are a shorthand way of expressing repeated multiplication. For example, 34=3×3×3×3=813^4 = 3 \times 3 \times 3 \times 3 = 8134=3×3×3×3=81. Let’s break down the rules for multiplying, dividing, and negative exponents step by step.

1. Multiplying Exponents

When you multiply powers with the same base, you add the exponents:

Rule:

am⋅an=am+na^m \cdot a^n = a^{m+n}am⋅an=am+n

Example:

23⋅24=23+4=27=1282^3 \cdot 2^4 = 2^{3+4} = 2^7 = 12823⋅24=23+4=27=128

If the bases are different but the exponents are the same, you multiply the bases and keep the exponent:

am⋅bm=(a⋅b)ma^m \cdot b^m = (a \cdot b)^mam⋅bm=(a⋅b)m

Example:

32⋅52=(3⋅5)2=152=2253^2 \cdot 5^2 = (3 \cdot 5)^2 = 15^2 = 22532⋅52=(3⋅5)2=152=225

2. Dividing Exponents

When you divide powers with the same base, you subtract the exponents:

Rule:

aman=am−n, where m>n\frac{a^m}{a^n} = a^{m-n}, \text{ where } m > nanam​=am−n, where m>n

Example:

5652=56−2=54=625\frac{5^6}{5^2} = 5^{6-2} = 5^4 = 6255256​=56−2=54=625

If m=nm = nm=n, the result is a0=1a^0 = 1a0=1 (explained more below).

If the bases are different but the exponents are the same:

ambm=(ab)m\frac{a^m}{b^m} = \left(\frac{a}{b}\right)^mbmam​=(ba​)m

Example:

6323=(62)3=33=27\frac{6^3}{2^3} = \left(\frac{6}{2}\right)^3 = 3^3 = 272363​=(26​)3=33=27

3. Negative Exponents

A negative exponent means you take the reciprocal of the base and make the exponent positive:

Rule:

a−m=1am, where a≠0a^{-m} = \frac{1}{a^m}, \text{ where } a \neq 0a−m=am1​, where a=0

Example:

2−3=123=182^{-3} = \frac{1}{2^3} = \frac{1}{8}2−3=231​=81​

If the base is already a fraction, flipping it makes the exponent positive:

Example:

(23)−2=(32)2=94\left(\frac{2}{3}\right)^{-2} = \left(\frac{3}{2}\right)^2 = \frac{9}{4}(32​)−2=(23​)2=49​

4. Special Cases

  1. Zero Exponent: Any base raised to the power of 0 is 1:

    a0=1, where a≠0a^0 = 1, \text{ where } a \neq 0a0=1, where a=0

    Example:

    70=17^0 = 170=1

  2. One as Exponent: Any base raised to the power of 1 is the base itself:

    a1=aa^1 = aa1=a

    Example:

    101=1010^1 = 10101=10

Summary of Rules

  • Multiplication: Add the exponents (am⋅an=am+na^m \cdot a^n = a^{m+n}am⋅an=am+n)

  • Division: Subtract the exponents (aman=am−n\frac{a^m}{a^n} = a^{m-n}anam​=am−n)

  • Negative Exponents: Flip the base (a−m=1ama^{-m} = \frac{1}{a^m}a−m=am1​)

  • Zero Exponent: Anything raised to 000 is 111 (a0=1a^0 = 1a0=1)

Practice Problems

Try these for yourself:

  1. Simplify: 35⋅323^5 \cdot 3^235⋅32

  2. Simplify: 104102\frac{10^4}{10^2}102104​

  3. Simplify: 2−32^{-3}2−3

  4. Simplify: (45)−2\left(\frac{4}{5}\right)^{-2}(54​)−2

Let me know if you’d like me to go over the answers! 😊