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

1. Simple Example:

   $$\left(\frac{3}{4}\right)^2 = \frac{3^2}{4^2} = \frac{9}{16}$$

2. Another Example:

   $$\left(\frac{2}{5}\right)^3 = \frac{2^3}{5^3} = \frac{8}{125}$$

3. 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}