Lesson 3: Simplifying Expressions Using the Product Property of Exponents

New Concept

An exponent shows repeated multiplication. To multiply powers that share the same base, you simply add their exponents together.

Product Property of Exponents
If $m$ and $n$ are real numbers and $x \neq 0$, then

What’s next

This is just the beginning! Next, you’ll master this property through worked examples and apply it to real-world problems involving massive numbers, like computer speeds.

Property

An exponent shows repeated multiplication. In an expression like $5^2$, the 5 is the base (the number being multiplied) and the 2 is the exponent (how many times to multiply it).

Examples

• $4^3 = 4 \cdot 4 \cdot 4 = 64$
• $(0.2)^4 = (0.2)(0.2)(0.2)(0.2) = 0.0016$
• $(\frac{1}{3})^5 = \frac{1}{3} \cdot \frac{1}{3} \cdot \frac{1}{3} \cdot \frac{1}{3} \cdot \frac{1}{3} = \frac{1}{243}$

Explanation

Think of exponents as a bossy little number telling the bigger number how many times to clone itself for a multiplication party! So, $4^3$ isn't $4 imes 3$, it's a party of three fours: $4 imes 4 imes 4$. It’s a super handy shortcut for writing out long, repetitive multiplications and looking like a math genius.

Property

If $m$ and $n$ are real numbers and $x \neq 0$, then $x^m \cdot x^n = x^{m+n}$.

Examples

• $b^4 \cdot b^8 \cdot b^2 = b^{4+8+2} = b^{14}$
• $p^3 \cdot p^6 \cdot q^5 \cdot q^9 = p^{3+6} \cdot q^{5+9} = p^9q^{14}$
• $10^4 \cdot 10^5 = 10^{4+5} = 10^9$

Explanation

When you multiply terms with the same base, just keep that base and add the exponents together! It's like combining two collections of the same thing. If you have a group of 3 apples ($a^3$) and another group of 5 apples ($a^5$), you don't magically get 15 apples; you just have a bigger group of 8 apples ($a^8$).