Power of a Fraction

Raising a fraction to a power follows a simple yet essential rule: the exponent is applied to both the numerator and the denominator.

If we have a fraction $ \frac{a}{b} $, where $ a $ is the numerator and $ b $ is the denominator, raising it to the power of $ n $ is expressed as: $$ \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} $$

Let’s dive into how this works, along with some special cases that might be tricky at first.

For instance, let’s calculate $ \left(\frac{2}{3}\right)^3 $. Using the rule, we get:

$$ \left(\frac{2}{3}\right)^3 = \frac{2^3}{3^3} = \frac{8}{27} $$

This rule holds true for any integer $ n $ (i.e., $ n \in \mathbb{Z} $).

That said, there are some exceptions and special cases worth noting:

Exponent Zero
Raising a fraction to the power of zero always equals 1, provided the numerator $ a $ is not zero: $$
\left(\frac{a}{b}\right)^0 = 1 \quad \text{if } a \neq 0 $$ For example: $$ \left(\frac{5}{7}\right)^0 = 1 $$ However, if $ a = 0 $ (i.e., the fraction is $ 0/b $), the rule doesn’t apply because $ 0^0 $ is undefined.
For example, with $$ \left(\frac{0}{2}\right)^0 = \frac{\color{red}{0^0}}{2^0} $$ we cannot compute the numerator.
Exponent One
Raising a fraction to the power of one leaves its value unchanged: $$ \left(\frac{a}{b}\right)^1 = \frac{a}{b} $$ For example: $$ \left(\frac{3}{4}\right)^1 = \frac{3^1}{4^1} = \frac{3}{4} $$

The same rules for manipulating exponents also apply to fractions.

Let’s break these rules down step by step to see how they work with fractions.

Multiplying Powers with the Same Base
When you multiply two fractions with the same base, you add the exponents: $$ \left(\frac{a}{b}\right)^m \cdot \left(\frac{a}{b}\right)^n = \left(\frac{a}{b}\right)^{m+n} $$ For example: $$ \left(\frac{2}{3}\right)^2 \cdot \left(\frac{2}{3}\right)^3 = \left(\frac{2}{3}\right)^{2+3} = \left(\frac{2}{3}\right)^5 $$
Dividing Powers with the Same Base
When you divide two fractions with the same base, you subtract the exponents: $$ \frac{\left(\frac{a}{b}\right)^m}{\left(\frac{a}{b}\right)^n} = \left(\frac{a}{b}\right)^{m-n} $$ For example: $$ \frac{\left(\frac{5}{4}\right)^6}{\left(\frac{5}{4}\right)^2} = \left(\frac{5}{4}\right)^{6-2} = \left(\frac{5}{4}\right)^4 $$
Raising a Power to Another Power
When a fraction raised to a power is raised again to another power, you multiply the exponents: $$ \left[\left(\frac{a}{b}\right)^m\right]^n = \left(\frac{a}{b}\right)^{m \cdot n} $$ For example: $$ \left[\left(\frac{3}{5}\right)^2\right]^4 = \left(\frac{3}{5}\right)^{2 \cdot 4} = \left(\frac{3}{5}\right)^8 $$
Multiplying Powers with the Same Exponent
If two fractions have the same exponent, you can multiply the bases and keep the shared exponent: $$
\left(\frac{a}{b}\right)^n \cdot \left(\frac{c}{d}\right)^n = \left(\frac{a \cdot c}{b \cdot d}\right)^n $$ For example: $$ \left(\frac{2}{3}\right)^3 \cdot \left(\frac{4}{5}\right)^3 = \left(\frac{2}{3} \cdot \frac{4}{5}\right)^3 = \left(\frac{2 \cdot 4}{3 \cdot 5}\right)^3 = \left(\frac{8}{15}\right)^3 $$
Dividing Powers with the Same Exponent
If two fractions share the same exponent, you can divide the bases and retain the common exponent: $$ \frac{\left(\frac{a}{b}\right)^n}{\left(\frac{c}{d}\right)^n} = \left(\frac{\frac{a}{b}}{\frac{c}{d}}\right)^n = \left(\frac{a \cdot d}{b \cdot c}\right)^n $$ For example: $$ \frac{\left(\frac{6}{7}\right)^2}{\left(\frac{2}{5}\right)^2} = \left(\frac{\frac{6}{7}}{\frac{2}{5}}\right)^2 = \left(\frac{6}{7} \cdot \frac{5}{2}\right)^2 = \left(\frac{6 \cdot 5}{7 \cdot 2}\right)^2 = \left(\frac{30}{14}\right)^2 = \left(\frac{15}{7}\right)^2 $$

These rules not only simplify calculations but also help you approach more complex problems involving fractions raised to powers with confidence.

They are fundamental to many areas of mathematics, from simplifying expressions to working with polynomials and exponential functions.

With consistent practice and a solid understanding of these principles, mastering powers of fractions becomes second nature.