Power of a Quotient Property
Grade 9 students in California Reveal Math Algebra 1 learn the Power of a Quotient Property: (a/b)^m = a^m/b^m where b≠0. The exponent distributes to both the numerator and denominator separately. After distributing, the Power of a Power rule may be needed — multiply exponents on variables. For example, (x/3)^4 = x^4/81, (2x^3/y^2)^3 = 8x^9/y^6, and (3a^2/4b^5)^2 = 9a^4/16b^10. A key caution: apply the exponent to every factor including coefficients, so nothing is left out.
Key Concepts
When a quotient (fraction) is raised to a power, distribute the exponent to both the numerator and the denominator separately:.
$$\left(\frac{a}{b}\right)^m = \frac{a^m}{b^m}, \quad b \neq 0$$.
Common Questions
What is the Power of a Quotient Property?
The Power of a Quotient Property states that (a/b)^m = a^m/b^m. When a fraction is raised to a power, you distribute that exponent to both the numerator and denominator separately.
How do you simplify (2x^3/y^2)^3?
Distribute the exponent 3 to every factor: 2^3=8, (x^3)^3=x^(3·3)=x^9, (y^2)^3=y^6. The result is 8x^9/y^6.
How do you simplify (3a^2/4b^5)^2?
Distribute exponent 2: 3^2=9, (a^2)^2=a^4, 4^2=16, (b^5)^2=b^10. The result is 9a^4/16b^10.
What is the Power of a Power rule used after distributing?
After distributing the outer exponent, multiply it by any existing exponents on variables. For (x^3)^3, multiply 3·3=9 to get x^9.
What common mistake should you avoid with this property?
Apply the exponent to every factor in both numerator and denominator, including coefficients. Missing any factor — like forgetting to raise the coefficient to the power — produces a wrong answer.
Which unit covers this property in Algebra 1?
This skill is from Unit 7: Exponents and Roots in California Reveal Math Algebra 1, Grade 9.