Algebra 2 / Unit 4: Radical Functions and Rational Exponents
4.3 Simplifying Radical Expressions
Notes
- Product rule: $\displaystyle \sqrt[n]{ab} = \sqrt[n]{a} \cdot \sqrt[n]{b}$. Pull out perfect powers: $\displaystyle \sqrt{50} = \sqrt{25 \cdot 2} = 5\sqrt{2}$ and $\displaystyle \sqrt[3]{54} = \sqrt[3]{27 \cdot 2} = 3\sqrt[3]{2}$.
- Like radicals (same index, same radicand) combine like terms: $\displaystyle 4\sqrt{3} + 2\sqrt{3} = 6\sqrt{3}$.
- Rationalize a denominator: $\displaystyle \frac{4}{\sqrt{2}} = \frac{4\sqrt{2}}{2} = 2\sqrt{2}$.
- For a binomial denominator, multiply by the conjugate: $\displaystyle (a + \sqrt{b})(a - \sqrt{b}) = a^2 - b$.
Practice
Simplify $\displaystyle \sqrt{72}$.
$72 = 36 \cdot 2$, so $\displaystyle \sqrt{72} = 6\sqrt{2}$.
Simplify $\displaystyle 3\sqrt{12} + \sqrt{27}$.
$\displaystyle 3\sqrt{12} = 6\sqrt{3}$ and $\displaystyle \sqrt{27} = 3\sqrt{3}$.
Rationalize $\displaystyle \frac{6}{\sqrt{3}}$.
$\displaystyle \frac{6\sqrt{3}}{3} = 2\sqrt{3}$.
Ready for more? Take the Unit 4 test.