Lesson 2: Properties of Rational Exponents and Radicals

Property

Powers with rational exponents obey the same laws of exponents as powers with integer exponents. For a base $a > 0$ and rational exponents $p$ and $q$:

1. First Law (Product of Powers): $a^p \cdot a^q = a^{p+q}$
2. Second Law (Quotient of Powers): $\frac{a^p}{a^q} = a^{p-q}$
3. Third Law (Power of a Power): $(a^p)^q = a^{pq}$
4. Fourth Law (Power of a Product): $(ab)^p = a^p b^p$

Examples

• To simplify $x^{1/2} \cdot x^{1/4}$, we add the exponents: $x^{1/2 + 1/4} = x^{2/4 + 1/4} = x^{3/4}$.

Property

We can multiply or divide radicals that have the same index, even if their radicands are different. The Product and Quotient Rules are used to combine the expressions under a single radical.

Product Rule:

Quotient Rule: