Product Property of Exponents
Apply the product property of exponents in Grade 9 algebra. Multiply expressions with the same base by adding exponents: xᵃ×xᵇ=x^(a+b) to simplify exponential expressions.
Key Concepts
Property The Product Property of Exponents states that the exponents of powers with the same base are added. $x^a \cdot x^b = x^{a+b}$.
Examples To multiply $y^5 \cdot y^3$, since the base 'y' is the same, we simply add the exponents: $y^{5+3} = y^8$. When negative exponents are involved, like in $z^6 \cdot z^{ 2}$, you still add them: $z^{6+( 2)} = z^4$. For expressions with coefficients like $3a^4 \cdot 2a^2$, multiply the coefficients and add the exponents: $(3 \cdot 2)a^{4+2} = 6a^6$.
Explanation When you multiply terms with the same base, you are just combining their powers into one. Instead of writing everything out, just add the exponents for a super fast shortcut. It’s like your exponents are teaming up! Remember, the base must be the same for this powerful trick to work. This property helps make very big problems much smaller.
Common Questions
What is the product property of exponents?
When multiplying powers with the same base, add the exponents: xᵃ × xᵇ = x^(a+b). For example, x³ × x⁵ = x⁸.
How do you apply the product property to multiple bases?
Handle each base separately. For 2x³ × 5x⁴, multiply coefficients (2 × 5 = 10) and add exponents of x (3 + 4 = 7) to get 10x⁷.
Why does multiplying powers with the same base mean adding exponents?
x³ means three factors of x; x⁵ means five factors. Together that is eight factors of x = x⁸. Adding exponents counts the total repeated factors.