Multiplying powers
Grade 8 math lesson on multiplying powers with the same base using the product rule for exponents. Students practice applying the rule that when multiplying expressions with the same base, the exponents are added together.
Key Concepts
Property When multiplying powers with the same base, add their exponents. $$x^a \cdot x^b = x^{a+b}$$.
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.
Common Questions
How do you multiply powers with the same base?
When multiplying powers with the same base, keep the base and add the exponents. The product rule states: a^m times a^n = a^(m+n). For example, 2^3 times 2^4 = 2^7.
Can you multiply powers with different bases?
You cannot directly combine powers with different bases unless you calculate each separately and then multiply the results. The product rule only works when both terms share the same base.
What is an example of multiplying powers?
Example: x^5 times x^3 = x^8, because 5 + 3 = 8. Another example: y^2 times y^7 = y^9. Always keep the base the same and just add the exponents.
Why do we add exponents when multiplying powers?
Adding exponents when multiplying works because of what exponents mean. x^3 means x times x times x (3 factors), and x^4 means 4 factors. Together they make 7 factors, so x^3 times x^4 = x^7.