Distributing with Variables and Exponents
Distribute terms with variables and exponents across parentheses in Grade 9 Algebra. Apply exponent rules while multiplying each term and then simplify the result.
Key Concepts
Property Use the Distributive Property with variables and remember to add exponents when multiplying powers with the same base: $x^a \cdot x^b = x^{a+b}$.
Examples $(9 y)8 = 8(9) + 8( y) = 72 8y$ $ab(ac + ad) = (ab)(ac) + (ab)(ad) = a^2bc + a^2bd$ $ yz(y^3 yz) = ( yz)(y^3) + ( yz)( yz) = y^4z + y^2z^2$.
Explanation Distributing works just the same with variables and letters. Multiply the term on the outside with each term on the inside. If you multiply variables with the same base, their exponents team up by adding their powers together! This makes simplifying long algebraic expressions a piece of cake.
Common Questions
How do you distribute a term with exponents across parentheses?
Multiply the outside term by each term inside the parentheses individually. Apply the product rule for exponents: xᵃ · xᵇ = xᵃ⁺ᵇ. Combine like terms after completing all the multiplications.
What exponent rule applies when distributing xⁿ · x^m?
The product rule states xⁿ · xᵐ = xⁿ⁺ᵐ—add the exponents when multiplying same-base terms. This rule must be applied correctly for every term inside the parentheses when distributing.
How does distributing with variables differ from distributing with numbers only?
When variables are present, you must track both the coefficient and the variable with its exponent. Multiply coefficients together as numbers and add exponents for matching bases, then combine any resulting like terms.