Product of Powers Property
Apply the product of powers property aᵐ·aⁿ=aᵐ⁺ⁿ in Grade 10 algebra to multiply expressions with the same base by adding exponents, simplifying both numerical and variable expressions.
Key Concepts
$$a^m \cdot a^n = a^{m+n}$$.
Simplify $x^2 \cdot x^5$: Since the base is the same, add the exponents: $x^2 \cdot x^5 = x^{2+5} = x^7$. Simplify $49^{\frac{1}{4}} \cdot 49^{\frac{3}{4}}$: $49^{\frac{1}{4}} \cdot 49^{\frac{3}{4}} = 49^{\frac{1}{4}+\frac{3}{4}} = 49^1 = 49$.
When you multiply two powers that share the same base, just keep the base and have a little party with the exponents by adding them together! This rule works for whole numbers and fractions alike. It’s a fantastic shortcut to combine expressions into one neat power, saving you from a lot of messy calculations and simplifying your work.
Common Questions
What is the product of powers property?
aᵐ · aⁿ = aᵐ⁺ⁿ. When multiplying powers with the same base, add the exponents. For example, x³·x⁵=x⁸ and 2⁴·2³=2⁷=128.
How do you simplify x⁴·x⁷ using the product of powers property?
Add exponents: x⁴·x⁷ = x⁴⁺⁷ = x¹¹.
Why must the bases be the same to apply the product of powers property?
The property only works when multiplying the same base: x²·x³=x⁵ but x²·y³ cannot be simplified this way because x and y are different bases.