Exponent Power of a Quotient Rule

The Exponent Power of a Quotient Rule tells you how to raise a fraction (or quotient) to an exponent. In mathematical terms, the rule is expressed as:

\[ \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} \]

where:

\(a\) is the numerator,
\(b\) is the denominator (and \(b \neq 0\)), and
\(n\) is any real number (often an integer).

How It Works

When you raise a fraction to a power, you are essentially multiplying the fraction by itself \(n\) times. For example:

\[ \left(\frac{a}{b}\right)^n = \underbrace{\frac{a}{b} \times \frac{a}{b} \times \cdots \times \frac{a}{b}}_{n \text{ times}} \]

Due to the associative property of multiplication, you can multiply all the numerators together and all the denominators together separately:

\[ \frac{a \times a \times \cdots \times a}{b \times b \times \cdots \times b} = \frac{a^n}{b^n} \]

Examples

Simple Example:
\[ \left(\frac{3}{4}\right)^2 = \frac{3^2}{4^2} = \frac{9}{16} \]
Another Example:
\[ \left(\frac{2}{5}\right)^3 = \frac{2^3}{5^3} = \frac{8}{125} \]
With a Negative Exponent:
A negative exponent means taking the reciprocal of the base raised to the positive exponent. For instance:
\[ \left(\frac{3}{4}\right)^{-2} = \frac{3^{-2}}{4^{-2}} = \frac{1/3^2}{1/4^2} = \frac{1/9}{1/16} = \frac{16}{9} \]
Alternatively, you can also think of it as:
\[ \left(\frac{3}{4}\right)^{-2} = \left(\frac{4}{3}\right)^2 = \frac{4^2}{3^2} = \frac{16}{9} \]

Key Points to Remember

Distribute the Exponent: The exponent applies to both the numerator and the denominator.
Denominator Non-Zero: Make sure the denominator \(b\) is not zero since division by zero is undefined.
Works for Various Exponents: This rule applies whether the exponent is positive, negative, or even a fractional exponent (with appropriate considerations for radicals).

Understanding this rule is fundamental for simplifying expressions in algebra, especially when working with fractions and exponents.

Latex

\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}