Lesson Objectives

- Demonstrate an understanding of radicals
- Learn how to identify like radicals
- Learn how to add radical expressions
- Learn how to subtract radical expressions

## Adding and Subtracting Radical Expressions

Now that we understand how to simplify radicals, we are ready to learn how to
add and subtract radicals. First and foremost, we must understand the concept of "like radicals". Like radicals have the same index and the same radicand. The
numbers multiplying the radicals can be different.

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: $$18x\sqrt{2x} - 20x\sqrt{2x} - 9x\sqrt{2x}$$ $$18x\sqrt{2x} - 20x\sqrt{2x} - 9x\sqrt{2x} = (18x - 20x - 9x)\sqrt{2x} = -11x\sqrt{2x}$$

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: $$18x\sqrt{2x} - 20x\sqrt{2x} - 9x\sqrt{2x}$$ $$18x\sqrt{2x} - 20x\sqrt{2x} - 9x\sqrt{2x} = (18x - 20x - 9x)\sqrt{2x} = -11x\sqrt{2x}$$

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