Integrated Math 2 / Unit 1: Exponents and Radicals
1.4 Simplifying Radicals
Notes
A radical is simplified when the radicand has no perfect-square factor (or perfect-cube factor for cube roots) left inside.
- $\displaystyle \sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$ for $a, b \geq 0$.
- Factor out the largest perfect power: $\displaystyle \sqrt{72} = \sqrt{36 \cdot 2} = 6\sqrt{2}$.
- Variables: $\displaystyle \sqrt{x^4} = x^2$ because $(x^2)^2 = x^4$.
- Cube roots: $\displaystyle \sqrt[3]{54} = \sqrt[3]{27 \cdot 2} = 3\sqrt[3]{2}$.
Practice
Simplify $\displaystyle \sqrt{72}$.
$72 = 36 \cdot 2$ and $\displaystyle \sqrt{36} = 6$. Choice $\displaystyle 2\sqrt{18}$ is not fully simplified.
Simplify $\displaystyle \sqrt{50x^4}$ for any real $x$.
$\displaystyle \sqrt{50} = 5\sqrt{2}$ and $\displaystyle \sqrt{x^4} = x^2$.
Simplify $\displaystyle \sqrt[3]{54}$.
$54 = 27 \cdot 2$ and $\displaystyle \sqrt[3]{27} = 3$.
Ready for more? Take the Unit 1 test.