Lesson Objectives
• Demonstrate an understanding of radicals
• Learn how to identify like radicals
Like Radicals: $$-12\sqrt{3}, 7\sqrt{3}$$ $$4\sqrt[3]{17}, 5\sqrt[3]{17}$$ $$13\sqrt[5]{22}, -8\sqrt[5]{22}$$ Not Like Radicals: $$5\sqrt{7}, 2\sqrt{3}$$ $$2\sqrt[4]{19}, 9\sqrt[4]{13}$$ $$-7\sqrt[5]{21}, -3\sqrt[3]{15}$$ We can combine "like radicals" using the distributive property. $$5\sqrt{2x} + 7\sqrt{2x} = (5 + 7)\sqrt{2x} = 12\sqrt{2x}$$ Let's look at a few examples.
Example 1: Simplify each $$4\sqrt{2} + 3\sqrt{2}$$ Since we have like radicals, we can perform operations with the numbers that are multiplying our radicals. $$4\sqrt{2} + 3\sqrt{2} = (4 + 3)\sqrt{2} = 7\sqrt{2}$$ Example 2: Simplify each $$2\sqrt{27} - \sqrt{3}$$ In this case, it appears that we do not have like radicals. When this scenario occurs, try to simplify each radical. $$2\sqrt{27} = 2 \cdot \sqrt{9} \cdot \sqrt{3} = 6\sqrt{3}$$ Now that we have simplified the first radical, we can see that we have like radicals. We will rewrite our problem as: $$6\sqrt{3} - \sqrt{3} = (6 - 1)\sqrt{3} = 5\sqrt{3}$$ Example 3: Simplify each $$-\sqrt[4]{486xy} - 4\sqrt[4]{96xy}$$ Let's first simplify each radical: $$-\sqrt[4]{486xy} = -\sqrt[4]{81} \cdot \sqrt[4]{6xy} = -3\sqrt[4]{6xy}$$ $$4\sqrt[4]{96xy} = 4\sqrt[4]{16} \cdot \sqrt[4]{6xy} = 8\sqrt[4]{6xy}$$ Now we can rewrite our problem as: $$-3\sqrt[4]{6xy} - 8\sqrt[4]{6xy}$$ $$-3\sqrt[4]{6xy} - 8\sqrt[4]{6xy} = (-3 - 8)\sqrt[4]{6xy} = -11\sqrt[4]{6xy}$$ Example 4: Simplify each $$3\sqrt{72x^3} - 5x\sqrt{32x} - 3\sqrt{18x^3}$$ Let's first simplify each radical: $$3\sqrt{72x^3} = 3 \cdot \sqrt{36x^2} \cdot \sqrt{2x} = 18x\sqrt{2x}$$ $$5x\sqrt{32x} = 5x \cdot \sqrt{16} \cdot \sqrt{2x} = 20x\sqrt{2x}$$ $$3\sqrt{18x^3} = 3 \cdot \sqrt{9x^2} \cdot \sqrt{2x} = 9x\sqrt{2x}$$ Now we can rewrite our problem as: $$18x\sqrt{2x} - 20x\sqrt{2x} - 9x\sqrt{2x}$$ $$18x\sqrt{2x} - 20x\sqrt{2x} - 9x\sqrt{2x} = (18x - 20x - 9x)\sqrt{2x} = -11x\sqrt{2x}$$