Power of a Quotient Property
Understand and apply Power of a Quotient Property in Grade 9 algebra. Use this rule to simplify, solve, and verify mathematical expressions efficiently.
Key Concepts
Property If $x$ and $y$ are any nonzero real numbers and $m$ is an integer, then $\left(\frac{x}{y}\right)^m = \frac{x^m}{y^m}$.
Examples To simplify $\left(\frac{3a^2}{4}\right)^3$, apply the exponent to the top and bottom: $\frac{(3a^2)^3}{4^3} = \frac{27a^6}{64}$. With negatives and variables: $\left(\frac{ 2x^3}{y^5}\right)^2 = \frac{( 2x^3)^2}{(y^5)^2} = \frac{4x^6}{y^{10}}$.
Explanation This rule is about fairness for fractions! Just like with products, an exponent outside a fraction's parentheses must be shared. It gets applied to both the numerator (the top part) and the denominator (the bottom part). This way, the entire fraction is properly raised to the power, keeping it balanced.
Common Questions
What is the Power of a Quotient Property?
The property states (a/b)ⁿ = aⁿ/bⁿ. When a fraction is raised to a power, apply the exponent to both the numerator and denominator.
How do you simplify (3/x)⁴ using the Power of a Quotient Property?
Apply the exponent to both parts: (3/x)⁴ = 3⁴/x⁴ = 81/x⁴.
Can the Power of a Quotient Property be combined with negative exponents?
Yes. Apply the exponent to numerator and denominator first, then use the negative exponent rule to rewrite or simplify further.