## Rules of Exponents Once you understand what an exponent is, the next step is learning how to **combine** them using consistent rules. These rules are used throughout electrical calculations, especially with: - Scientific notation - Power formulas - Unit conversions - Root and square power operations These exponent rules help you **simplify** and **solve** expressions quickly and correctly. ### 1. Product Rule **When multiplying powers with the same base, add the exponents:** $ a^m \cdot a^n = a^{m+n} $ **Example:** $ 10^3 \cdot 10^2 = 10^{3+2} = 10^5 = 100000 $ ### 2. Quotient Rule **When dividing powers with the same base, subtract the exponents:** $ \frac{a^m}{a^n} = a^{m-n} $ **Example:** $ \frac{10^5}{10^2} = 10^{5-2} = 10^3 = 1000 $ ### 3. Power of a Power Rule **When raising a power to another power, multiply the exponents:** $ (a^m)^n = a^{m \cdot n} $ **Example:** $ (10^2)^3 = 10^{2 \cdot 3} = 10^6 = 1000000 $ ### 4. Power of a Product Rule **When raising a product to a power, apply the exponent to each factor:** $ (ab)^n = a^n \cdot b^n $ **Example:** $ (2 \cdot 10)^3 = 2^3 \cdot 10^3 = 8 \cdot 1000 = 8000 $ ### 5. Zero Exponent Rule **Any nonzero number raised to the 0 power is 1:** $ a^0 = 1 \quad (\text{as long as } a \neq 0) $ **Examples:** $ 10^0 = 1,\quad 5^0 = 1 $ ### 6. Negative Exponent Rule **A negative exponent means to take the reciprocal:** $ a^{-n} = \frac{1}{a^n} $ **Examples:** $ 10^{-2} = \frac{1}{10^2} = \frac{1}{100} = 0.01 $ $ 2^{-3} = \frac{1}{2^3} = \frac{1}{8} $ ### Electrical Examples |Expression|Meaning|Result| |---|---|---| |$10^6 \cdot 10^3$|Multiply powers of 10|$10^9 = 1\ \text{G}$| |$\frac{10^6}{10^3}$|Divide MΩ by kΩ|$10^3 = 1\ \text{k}$| |$(10^3)^2$|kV squared|$10^6 = 1\ \text{MV}$| |$4.7 \cdot 10^3$|4.7 kΩ|$4700\ \Omega$| |$2.0 \cdot 10^{-6}$|2.0 μF|$0.000002\ \text{F}$| ### Practice Simplify the following using exponent rules: 1. $10^4 \cdot 10^2 =\ ?$ 2. $\frac{10^6}{10^3} =\ ?$ 3. $(10^2)^3 =\ ?$ 4. $(2 \cdot 10)^2 =\ ?$ 5. $10^{-2} =\ ?$ ## Summary - Add exponents when multiplying, subtract when dividing - Multiply exponents when raising a power to a power - Any nonzero number to the 0 power is 1 - Negative exponents flip the number into the denominator - These rules help simplify and evaluate common electrical expressions involving units, scale, and scientific notation