Products of variables
Products of Variables explains how to multiply variable expressions by using the commutative and associative properties to rearrange and group factors, then applying the first law of exponents to combine like variables. Covered in Yoshiwara Elementary Algebra Chapter 5: Exponents and Roots, Grade 6 students learn to simplify products like (3b)(4b) = 12b² by multiplying numerical coefficients and adding exponents on matching variables. This skill is foundational for polynomial multiplication and simplification.
Key Concepts
Property The commutative and associative properties tell us that we can multiply the factors of a product in any order. To simplify a product like $(3b)(4b)$, we can rearrange and group the factors: $$(3b)(4b) = (3 \cdot 4) \cdot (b \cdot b) = 12b^2$$.
Examples To simplify $(6a)( 4a)$, we multiply the coefficients and the variables separately: $(6)( 4) \cdot (a)(a) = 24a^2$.
To simplify $(3x)^3$, we multiply three copies of the expression: $(3x)(3x)(3x) = (3 \cdot 3 \cdot 3) \cdot (x \cdot x \cdot x) = 27x^3$.
Common Questions
How do you multiply variable expressions?
Rearrange the factors using the commutative and associative properties to group coefficients together and matching variables together. Multiply the numbers and add exponents for matching variables.
What is (3b)(4b)?
Multiply the coefficients: 3 × 4 = 12. Then apply the exponent rule to the variables: b × b = b². The result is 12b².
What rule applies to variables when you multiply them?
The first law of exponents: a^m × a^n = a^(m+n). When multiplying variables with the same base, add the exponents.
Where are products of variables covered in Yoshiwara Elementary Algebra?
This concept is in Chapter 5: Exponents and Roots of Yoshiwara Elementary Algebra.
How do you multiply expressions with more than one variable, like (2x²y)(3xy³)?
Group like variables: (2)(3) × (x²·x) × (y·y³) = 6 × x³ × y⁴ = 6x³y⁴.