Adding Radicals Lesson

Like Radicals

Before we can begin adding and subtracting radicals, we need to understand the term "like radicals". Like radicals are radicals that have the same index and the same radicand. Let's look at a few examples:
Like Radicals: $$2\sqrt{5},\hspace{.5em}3\sqrt{5}$$ In each case, our radical is the square root of 5. The index is 2 and the radicand is 5. The numbers multiplying the radical do not matter. $$-7\sqrt[5]{13},\hspace{.5em}20\sqrt[5]{13}$$ In each case, our radical is the fifth root of 13. The index is 5 and the radicand is 13. The numbers multiplying the radical do not matter.
Not Like Radicals: $$4\sqrt{7}, \hspace{.5em}-3\sqrt[3]{7}$$ We do not have like radicals, although the radicand (7) is the same in each case, the index is different (2 vs 3). $$-8\sqrt[3]{12}, \hspace{.5em}2\sqrt[3]{19}$$ We do not have like radicals, although the index (3) is the same in each case, the radicands are different (12 vs 19).

Adding & Subtracting Radicals

To add or subtract like radicals, we perform operations with the numbers multiplying the radicals and leave the radical part unchanged. Let's look at a few examples.
Example 1: Perform each indicated operation. $$7\sqrt{5}+ 3\sqrt{5}$$ We will add 7 and 3, the radical (square root of 5) remains unchanged: $$7\sqrt{5}+ 3\sqrt{5}=$$ $$(7 + 3)\sqrt{5}=10\sqrt{5}$$ Example 2: Perform each indicated operation. $$-2\sqrt[5]{9}- 3\sqrt[5]{9}$$ We will subtract -2 - 3, the radical (fifth root of 9) remains unchanged: $$-2\sqrt[5]{9}- 3\sqrt[5]{9}=$$ $$(-2 - 3)\sqrt[5]{9}=-5\sqrt[5]{9}$$ In some cases, we may not appear to have like radicals. We need to simplify each radical first and then determine if we have like radicals.
Example 3: Perform each indicated operation. $$2\sqrt{24}- 2\sqrt{54}$$ At this point, it seems as though we do not have like radicals. Let's simplify each radical: $$2\sqrt{24}=2 \cdot \sqrt{4}\cdot \sqrt{6}=4\sqrt{6}$$ $$2\sqrt{54}=2 \cdot \sqrt{9}\cdot \sqrt{6}=6\sqrt{6}$$ We can rewrite our problem as: $$2\sqrt{24}- 2\sqrt{54}=4\sqrt{6}- 6\sqrt{6}$$ Now that we have like radicals, we can perform our subtraction: $$4\sqrt{6}- 6\sqrt{6}=-2\sqrt{6}$$ Example 4: Perform each indicated operation. $$-4\sqrt[3]{6}- \sqrt[3]{162}+ 2\sqrt[3]{-24}$$ Let's first simplify each part: $$-4\sqrt[3]{6}$$ The first part is already simplified, let's look at the next part: $$\sqrt[3]{162}$$ Factoring 162 yields:
162 = 3 • 3 • 3 • 3 • 2
Since we have a cube root, we are looking for a perfect cube. We have a factor of 27 (3 factors of 3), which can be used to simplify: $$\sqrt[3]{162}=\sqrt[3]{27}\cdot \sqrt[3]{6}$$ $$\sqrt[3]{27}\cdot \sqrt[3]{6}=3\sqrt[3]{6}$$ Lastly, let's look at: $$2\sqrt[3]{-24}$$ Let's think about -24, we can factor this as:
-24 = -1 • 2 • 2 • 2 • 3
-1 is a perfect cube:
-1 = -1 • -1 • -1
8 is also a perfect cube:
8 = 2 • 2 • 2
We can use both to simplify: $$2\sqrt[3]{-24}=2 \cdot \sqrt[3]{-1}\cdot \sqrt[3]{8}\cdot \sqrt[3]{3}=$$ $$2 \cdot -1 \cdot 2 \cdot \sqrt[3]{3}=-4\sqrt[3]{3}$$ We can rewrite our problem as: $$-4\sqrt[3]{6}- 3\sqrt[3]{6}- 4\sqrt[3]{3}$$ We have two like radicals and one radical which can't be combined with anything. $$-4\sqrt[3]{6}- 3\sqrt[3]{6}- 4\sqrt[3]{3}=$$ $$-7\sqrt[3]{6}- 4\sqrt[3]{3}$$

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