Power of a Quotient
The Power of a Quotient rule states that to raise a fraction to a power, you raise both the numerator and denominator separately to that power: (a/b)^n = a^n/b^n. This exponent property is taught in Yoshiwara Elementary Algebra Chapter 9: More About Exponents and Roots and is fundamental for Grade 6 students simplifying rational expressions and algebraic fractions. Applying this rule correctly prevents common errors when working with fractional bases raised to exponents.
Key Concepts
Property To raise a quotient to a power, raise both the numerator and the denominator to the power. In symbols, $$\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$$.
Examples To simplify $(\frac{a}{3})^4$, raise both the numerator and denominator to the fourth power: $\frac{a^4}{3^4} = \frac{a^4}{81}$. For $(\frac{x^2}{y^3})^2$, apply the outer exponent to both parts and simplify: $\frac{(x^2)^2}{(y^3)^2} = \frac{x^4}{y^6}$. Simplify $(\frac{ 2}{z})^3$ by applying the exponent to the top and bottom: $\frac{( 2)^3}{z^3} = \frac{ 8}{z^3}$.
Explanation This is like distributing the exponent to a fraction. Raising a fraction to a power means multiplying the fraction by itself that many times. This results in the numerator and denominator each being raised to that power.
Common Questions
What is the power of a quotient rule?
The power of a quotient rule states (a/b)^n = a^n / b^n — raise both the numerator and the denominator to the exponent separately.
How do you apply the power of a quotient?
Distribute the exponent to both the top and bottom of the fraction. For example, (2/3)^3 = 2^3 / 3^3 = 8/27.
Where is the power of a quotient taught in Yoshiwara Elementary Algebra?
It is covered in Chapter 9: More About Exponents and Roots in Yoshiwara Elementary Algebra, as part of a larger set of exponent laws.
Can you use the power of a quotient rule with variables?
Yes. For example, (x/y)^4 = x^4/y^4. This rule works with any non-zero base expressions.
What mistakes do students make with the power of a quotient?
A common mistake is only applying the exponent to the numerator and forgetting the denominator. Always raise both parts of the fraction to the given power.