Lesson 2: Properties of 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

For any positive real numbers $a$ and $b$, and positive integer $n \geq 2$:

Product Property: $\sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{ab}$

Property

A radical expression is considered simplified if there are

• no factors in the radicand have perfect powers of the index
• no fractions in the radicand
• no radicals in the denominator of a fraction

Examples

• Is $\sqrt{50}$ simplified? No, because $\sqrt{50} = \sqrt{25 \cdot 2} = 5\sqrt{2}$. It contained a perfect square factor.
• Is $\sqrt{\dfrac{5}{9}}$ simplified? No, because it has a fraction in the radicand. It simplifies to $\dfrac{\sqrt{5}}{\sqrt{9}} = \dfrac{\sqrt{5}}{3}$.
• Is $\dfrac{7}{\sqrt{3}}$ simplified? No, because there is a radical in the denominator. It must be rationalized to become $\dfrac{7\sqrt{3}}{3}$.

Explanation

A radical is fully simplified when it's completely tidy. This means no perfect squares (or cubes, etc.) are left inside, no fractions are under the radical sign, and no radicals are hiding in the denominator of a fraction.