Lesson 27: Laws of Exponents

New Concept

The Laws of Exponents are rules that simplify expressions involving powers with the same base. Here are the three fundamental laws:

• $x^a \cdot x^b = x^{a+b}$
• $\frac{x^a}{x^b} = x^{a-b} \quad \text{for} x≠0$
• $(x^a)^b = x^{ab}$

What’s next

Next, we'll break down these laws with worked examples for multiplication, division, and raising a power to a power, building your problem-solving speed.

Property

When multiplying powers with the same base, add their exponents.

Examples

• To simplify $z^4 \cdot z^3$, we add the exponents: $z^{4+3} = z^7$.
• With numbers, it's the same: $5^2 \cdot 5^5 = 5^{2+5} = 5^7$.
• Even with a variable by itself: $y \cdot y^6 = y^1 \cdot y^6 = y^{1+6} = y^7$.

Explanation

Think of this as combining teams! When you multiply powers that share the same base, you are just adding all the factors together into one big group. Instead of counting every single factor, you can simply add the exponents to find the total size of your new super-team. It's a quick way to handle big numbers.

Property

When dividing powers with the same base, subtract the exponent of the denominator from the exponent of the numerator.

Examples

• To simplify $\frac{4^8}{4^5}$, we subtract the exponents: $4^{8-5} = 4^3$.
• For variables, it's identical: $\frac{m^{10}}{m^6} = m^{10-6} = m^4$ (as long as $m \neq 0$).
• A bigger example: $\frac{10^{15}}{10^9} = 10^{15-9} = 10^6$.

Explanation

Imagine a battle where factors from the top cancel out factors from the bottom. This rule is a shortcut to see who wins! By subtracting the exponents, you find out how many factors are left over and where they are. Remember, the base can't be zero because dividing by zero is a universal math foul that breaks everything!