The Product Rule of Exponents
Apply the product rule of exponents (a^m · a^n = a^(m+n)) to multiply expressions with the same base by adding their exponents in Grade 9 Algebra.
Key Concepts
Property For any real number $a$ and natural numbers $m$ and $n$, the product rule of exponents states that $$a^m \cdot a^n = a^{m+n}$$.
Examples To simplify $g^4 \cdot g^2$, we add the exponents: $g^{4+2} = g^6$. To simplify $( 5)^4 \cdot ( 5)$, we recognize that $( 5)$ is $( 5)^1$. So, we have $( 5)^{4+1} = ( 5)^5$. We can combine multiple terms: $y^3 \cdot y^6 \cdot y^2 = y^{3+6+2} = y^{11}$.
Explanation When multiplying terms with the same base, keep the base and add the exponents. This is a shortcut for counting all the individual factors. For example, $x^2 \cdot x^3$ means $(x \cdot x) \cdot (x \cdot x \cdot x)$, which is simply $x^5$.
Common Questions
What is the product rule of exponents?
The product rule states that when multiplying two powers with the same base, you add the exponents: aᵐ × aⁿ = aᵐ⁺ⁿ. For example, x³ × x⁴ = x⁷. The base stays the same — only the exponents are combined through addition.
Why do you add exponents when multiplying same-base powers?
Exponents represent repeated multiplication. x³ means x × x × x and x⁴ means x × x × x × x. Multiplying them gives x × x × x × x × x × x × x, which equals x⁷. Adding exponents (3 + 4 = 7) is a shortcut for counting the total number of multiplied factors.
Does the product rule apply to different bases?
No. The product rule only applies when the bases are identical. You can simplify x³ × x⁴ = x⁷, but you cannot simplify x³ × y⁴ further because x and y are different bases. For different bases, you can only combine them if they have the same exponent using (xy)ⁿ = xⁿyⁿ.