Properties of Exponents and Radicals

Working with exponents becomes much easier when you know the key rules. When multiplying powers with the same base, use the Product Rule by adding exponents: $a^x \times a^y = a^{x+y}$. For division, the Quotient Rule tells us to subtract exponents: $a^x \div a^y = a^{x-y}$.

Need to raise a power to another power? The Power Rule simplifies this by multiplying exponents: $(a^x)^y = a^{xy}$. When dealing with products or fractions raised to a power, remember that the Power of a Product Rule gives us $(ab)^x = a^xb^x$ and the Power of a Fraction Rule gives us $(\frac{a}{b})^x = \frac{a^x}{b^x}$.

Any number raised to the zero power equals 1 ($a^0 = 1$), while negative exponents flip the expression to the denominator: $a^{-x} = \frac{1}{a^x}$. For radicals, the Product Property allows us to separate roots: $\sqrt[x]{ab} = \sqrt[x]{a} \cdot \sqrt[x]{b}$, and the Quotient Property works similarly: $\sqrt[x]{a/b} = \sqrt[x]{a} / \sqrt[x]{b}$.

Pro Tip: When dealing with even-numbered roots like square roots, remember that $\sqrt[x]{a^x} = |a|$ (the absolute value), but for odd-numbered roots, $\sqrt[x]{a^x} = a$ (no absolute value needed).