Adding and Subtracting Radical Expressions Worksheet for 8th - 9th ... - Free Printable

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Adding and Subtracting Radical Expressions

Simplify:

1. $ 5\sqrt{6} + 4\sqrt{6} $

2. $ 3\sqrt{7} + 6\sqrt{7} $

3. $ -11\sqrt{12} + 14\sqrt{12} $

4. $ -6\sqrt{13} + 9\sqrt{13} $

5. $ 10\sqrt{11} + 17\sqrt{11} $

6. $ -7\sqrt{17} + 8\sqrt{17} $

7. $ 9\sqrt{8} - 2\sqrt{8} $

8. $ 2\sqrt{5} + 2\sqrt{5} $

9. $ -8\sqrt{2} + 2\sqrt{2} $

10. $ 3\sqrt{3} - \sqrt{27} $

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Let’s solve each problem step by step and explain how to add or subtract radical expressions.

🔹 Key Rule:

You can only combine radical terms if they have the same radicand (the number under the square root) and the same index (here, all are square roots).

So:
$$
a\sqrt{b} + c\sqrt{b} = (a + c)\sqrt{b}
$$

If the radicals are not like terms (different radicands), you cannot combine them.

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✔ Let's Solve Each Problem:

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1. $ 5\sqrt{6} + 4\sqrt{6} $

Same radicand: $ \sqrt{6} $

Add coefficients: $ 5 + 4 = 9 $

Answer: $ \boxed{9\sqrt{6}} $

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2. $ 3\sqrt{7} + 6\sqrt{7} $

Same radicand: $ \sqrt{7} $

$ 3 + 6 = 9 $

Answer: $ \boxed{9\sqrt{7}} $

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3. $ -11\sqrt{12} + 14\sqrt{12} $

Same radicand: $ \sqrt{12} $

$ -11 + 14 = 3 $

Answer: $ \boxed{3\sqrt{12}} $

But we can simplify $ \sqrt{12} $ further:
$$
\sqrt{12} = \sqrt{4 \cdot 3} = 2\sqrt{3}
$$

So:
$$
3\sqrt{12} = 3 \cdot 2\sqrt{3} = \boxed{6\sqrt{3}}
$$

✔ Final answer: $ \boxed{6\sqrt{3}} $

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4. $ -6\sqrt{13} + 9\sqrt{13} $

Same radicand: $ \sqrt{13} $

$ -6 + 9 = 3 $

Answer: $ \boxed{3\sqrt{13}} $

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5. $ 10\sqrt{11} + 17\sqrt{11} $

Same radicand: $ \sqrt{11} $

$ 10 + 17 = 27 $

Answer: $ \boxed{27\sqrt{11}} $

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6. $ -7\sqrt{17} + 8\sqrt{17} $

Same radicand: $ \sqrt{17} $

$ -7 + 8 = 1 $

Answer: $ \boxed{\sqrt{17}} $

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7. $ 9\sqrt{8} - 2\sqrt{8} $

Same radicand: $ \sqrt{8} $

$ 9 - 2 = 7 $

Answer: $ \boxed{7\sqrt{8}} $

Now simplify $ \sqrt{8} $:
$$
\sqrt{8} = \sqrt{4 \cdot 2} = 2\sqrt{2}
$$

So:
$$
7\sqrt{8} = 7 \cdot 2\sqrt{2} = \boxed{14\sqrt{2}}
$$

✔ Final answer: $ \boxed{14\sqrt{2}} $

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8. $ 2\sqrt{5} + 2\sqrt{5} $

Same radicand: $ \sqrt{5} $

$ 2 + 2 = 4 $

Answer: $ \boxed{4\sqrt{5}} $

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9. $ -8\sqrt{2} + 2\sqrt{2} $

Same radicand: $ \sqrt{2} $

$ -8 + 2 = -6 $

Answer: $ \boxed{-6\sqrt{2}} $

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10. $ 3\sqrt{3} - \sqrt{27} $

First, simplify $ \sqrt{27} $:
$$
\sqrt{27} = \sqrt{9 \cdot 3} = 3\sqrt{3}
$$

Now substitute:
$$
3\sqrt{3} - 3\sqrt{3} = 0
$$

Answer: $ \boxed{0} $

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✔ Final Answers:

1. $ 9\sqrt{6} $
2. $ 9\sqrt{7} $
3. $ 6\sqrt{3} $
4. $ 3\sqrt{13} $
5. $ 27\sqrt{11} $
6. $ \sqrt{17} $
7. $ 14\sqrt{2} $
8. $ 4\sqrt{5} $
9. $ -6\sqrt{2} $
10. $ 0 $

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🔍 Summary of Steps:

- Combine like radicals (same radicand).
- Simplify radicals first if needed.
- Use distributive property: $ a\sqrt{b} + c\sqrt{b} = (a+c)\sqrt{b} $
- Always check if the radical can be simplified before or after combining.

Let me know if you'd like this formatted as a printable PDF or want explanations for more complex problems!