Adding And Subtracting Radical Expressions Worksheet, Examples ... - Free Printable

Let's solve each problem step by step, simplifying the radical expressions. The key idea is that we can only combine radicals if they are "like radicals", meaning they have the same radicand (the number under the square root). If not, we must first simplify each radical to see if they can be combined.

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1) $ 2\sqrt{2} + 3\sqrt{2} $

Both terms have $ \sqrt{2} $, so they are like radicals.

Add coefficients:
$$
2 + 3 = 5
$$

So,
$$
2\sqrt{2} + 3\sqrt{2} = 5\sqrt{2}
$$

✔ Answer: $ 5\sqrt{2} $

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2) $ 6\sqrt{7} - 4\sqrt{7} $

Like radicals again (both have $ \sqrt{7} $).

Subtract coefficients:
$$
6 - 4 = 2
$$

So,
$$
6\sqrt{7} - 4\sqrt{7} = 2\sqrt{7}
$$

✔ Answer: $ 2\sqrt{7} $

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3) $ \sqrt{50} - \sqrt{2} $

First, simplify $ \sqrt{50} $:

$$
\sqrt{50} = \sqrt{25 \cdot 2} = \sqrt{25} \cdot \sqrt{2} = 5\sqrt{2}
$$

Now substitute:
$$
5\sqrt{2} - \sqrt{2} = (5 - 1)\sqrt{2} = 4\sqrt{2}
$$

✔ Answer: $ 4\sqrt{2} $

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4) $ \sqrt{10} + \sqrt{40} $

Simplify $ \sqrt{40} $:

$$
\sqrt{40} = \sqrt{4 \cdot 10} = \sqrt{4} \cdot \sqrt{10} = 2\sqrt{10}
$$

Now:
$$
\sqrt{10} + 2\sqrt{10} = (1 + 2)\sqrt{10} = 3\sqrt{10}
$$

✔ Answer: $ 3\sqrt{10} $

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5) $ \sqrt{63} + \sqrt{12} $

Simplify each radical:

- $ \sqrt{63} = \sqrt{9 \cdot 7} = \sqrt{9} \cdot \sqrt{7} = 3\sqrt{7} $
- $ \sqrt{12} = \sqrt{4 \cdot 3} = \sqrt{4} \cdot \sqrt{3} = 2\sqrt{3} $

Now:
$$
3\sqrt{7} + 2\sqrt{3}
$$

These are not like radicals (different radicands), so they cannot be combined.

✔ Answer: $ 3\sqrt{7} + 2\sqrt{3} $

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6) $ \sqrt{32} - \sqrt{20} $

Simplify each:

- $ \sqrt{32} = \sqrt{16 \cdot 2} = \sqrt{16} \cdot \sqrt{2} = 4\sqrt{2} $
- $ \sqrt{20} = \sqrt{4 \cdot 5} = \sqrt{4} \cdot \sqrt{5} = 2\sqrt{5} $

Now:
$$
4\sqrt{2} - 2\sqrt{5}
$$

Different radicands → cannot combine.

✔ Answer: $ 4\sqrt{2} - 2\sqrt{5} $

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7) $ \sqrt{24} - \sqrt{6} $

Simplify $ \sqrt{24} $:

$$
\sqrt{24} = \sqrt{4 \cdot 6} = \sqrt{4} \cdot \sqrt{6} = 2\sqrt{6}
$$

Now:
$$
2\sqrt{6} - \sqrt{6} = (2 - 1)\sqrt{6} = \sqrt{6}
$$

✔ Answer: $ \sqrt{6} $

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8) $ \sqrt{42} + \sqrt{14} $

Check if either can be simplified:

- $ \sqrt{42} = \sqrt{6 \cdot 7} $ → no perfect squares → stays as $ \sqrt{42} $
- $ \sqrt{14} = \sqrt{2 \cdot 7} $ → also no perfect squares → stays as $ \sqrt{14} $

No simplification possible, and radicands are different → cannot combine.

✔ Answer: $ \sqrt{42} + \sqrt{14} $

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✔ Final Answers:

1) $ 5\sqrt{2} $
2) $ 2\sqrt{7} $
3) $ 4\sqrt{2} $
4) $ 3\sqrt{10} $
5) $ 3\sqrt{7} + 2\sqrt{3} $
6) $ 4\sqrt{2} - 2\sqrt{5} $
7) $ \sqrt{6} $
8) $ \sqrt{42} + \sqrt{14} $

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🔍 Key Concepts Recap:

- Like radicals: Same radicand (e.g., $ \sqrt{2}, 3\sqrt{2} $) → can add/subtract.
- Simplify radicals first by factoring out perfect squares.
- Only combine like terms after simplification.

Let me know if you'd like a version with steps written out for printing!