Power of a Product Property
This Grade 6 algebra skill from Yoshiwara Elementary Algebra introduces the power of a product property for exponents: (ab)^n = a^n x b^n. Students learn to distribute an exponent to each factor inside parentheses and apply this rule alongside other exponent laws to simplify expressions.
Key Concepts
Property To raise a product to a power, raise each factor to the power. In symbols, $$(ab)^n = a^nb^n$$.
Examples To simplify $(4xy)^2$, apply the exponent to each factor inside: $4^2x^2y^2 = 16x^2y^2$. For $( 3a^2)^3$, raise each factor to the third power: $( 3)^3(a^2)^3 = 27a^6$. Note the difference: in $5x^3$, only $x$ is cubed. In $(5x)^3$, both 5 and $x$ are cubed, giving $125x^3$.
Explanation This rule works because multiplication is commutative. An expression like $(2x)^3$ means $(2x)(2x)(2x)$. You can regroup the factors as $(2 \cdot 2 \cdot 2)(x \cdot x \cdot x)$, which is simply $2^3x^3$.
Common Questions
What is the power of a product property?
The power of a product property states (ab)^n = a^n x b^n. The exponent applies to each factor in the product.
How do you apply the power of a product property?
Raise each factor inside the parentheses to the given exponent. For example, (2x)^3 = 2^3 x x^3 = 8x^3.
Does the power of a product property work with more than two factors?
Yes. (abc)^n = a^n x b^n x c^n. The exponent distributes to every factor.
What is a common mistake when applying this property?
Students sometimes forget to raise the coefficient to the power. For example, (3x)^2 = 9x^2, not 3x^2.
Where is the power of a product property taught?
The power of a product property is covered in the Yoshiwara Elementary Algebra textbook for Grade 6.