Algebra 1 worksheet focusing on adding and subtracting radicals, featuring 25 problems with various radical expressions to simplify.

To solve the problems involving adding and subtracting radicals, we need to simplify each expression by combining like terms. Like terms are those that have the same radical part. Here's how to approach each problem step by step:

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General Steps:

1. Simplify each radical term if possible (e.g., $\sqrt{40} = \sqrt{4 \cdot 10} = 2\sqrt{10}$).
2. Combine like terms (terms with the same radical part).

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Solutions:

#### 1) $ 2\sqrt{6} - 3\sqrt{6} $
- Both terms have the same radical part ($\sqrt{6}$).
- Combine the coefficients: $2 - 3 = -1$.
- Result: $-\sqrt{6}$.

$$
\boxed{-\sqrt{6}}
$$

#### 2) $ -5\sqrt{7} + 2\sqrt{7} $
- Both terms have the same radical part ($\sqrt{7}$).
- Combine the coefficients: $-5 + 2 = -3$.
- Result: $-3\sqrt{7}$.

$$
\boxed{-3\sqrt{7}}
$$

#### 3) $ -3\sqrt{2} - 4\sqrt{2} $
- Both terms have the same radical part ($\sqrt{2}$).
- Combine the coefficients: $-3 - 4 = -7$.
- Result: $-7\sqrt{2}$.

$$
\boxed{-7\sqrt{2}}
$$

#### 4) $ -2\sqrt{3} + 2\sqrt{3} $
- Both terms have the same radical part ($\sqrt{3}$).
- Combine the coefficients: $-2 + 2 = 0$.
- Result: $0$.

$$
\boxed{0}
$$

#### 5) $ -3\sqrt{6} - 5\sqrt{6} $
- Both terms have the same radical part ($\sqrt{6}$).
- Combine the coefficients: $-3 - 5 = -8$.
- Result: $-8\sqrt{6}$.

$$
\boxed{-8\sqrt{6}}
$$

#### 6) $ -2\sqrt{40} + 5\sqrt{10} $
- Simplify $\sqrt{40}$: $\sqrt{40} = \sqrt{4 \cdot 10} = 2\sqrt{10}$.
- Substitute: $-2\sqrt{40} = -2(2\sqrt{10}) = -4\sqrt{10}$.
- Now the expression is: $-4\sqrt{10} + 5\sqrt{10}$.
- Combine the coefficients: $-4 + 5 = 1$.
- Result: $\sqrt{10}$.

$$
\boxed{\sqrt{10}}
$$

#### 7) $ -\sqrt{8} - 4\sqrt{200} $
- Simplify $\sqrt{8}$: $\sqrt{8} = \sqrt{4 \cdot 2} = 2\sqrt{2}$.
- Simplify $\sqrt{200}$: $\sqrt{200} = \sqrt{100 \cdot 2} = 10\sqrt{2}$.
- Substitute: $-\sqrt{8} = -2\sqrt{2}$ and $-4\sqrt{200} = -4(10\sqrt{2}) = -40\sqrt{2}$.
- Now the expression is: $-2\sqrt{2} - 40\sqrt{2}$.
- Combine the coefficients: $-2 - 40 = -42$.
- Result: $-42\sqrt{2}$.

$$
\boxed{-42\sqrt{2}}
$$

#### 8) $ -2\sqrt{80} + 2\sqrt{45} $
- Simplify $\sqrt{80}$: $\sqrt{80} = \sqrt{16 \cdot 5} = 4\sqrt{5}$.
- Simplify $\sqrt{45}$: $\sqrt{45} = \sqrt{9 \cdot 5} = 3\sqrt{5}$.
- Substitute: $-2\sqrt{80} = -2(4\sqrt{5}) = -8\sqrt{5}$ and $2\sqrt{45} = 2(3\sqrt{5}) = 6\sqrt{5}$.
- Now the expression is: $-8\sqrt{5} + 6\sqrt{5}$.
- Combine the coefficients: $-8 + 6 = -2$.
- Result: $-2\sqrt{5}$.

$$
\boxed{-2\sqrt{5}}
$$

#### 9) $ -2\sqrt{28} - 3\sqrt{112} $
- Simplify $\sqrt{28}$: $\sqrt{28} = \sqrt{4 \cdot 7} = 2\sqrt{7}$.
- Simplify $\sqrt{112}$: $\sqrt{112} = \sqrt{16 \cdot 7} = 4\sqrt{7}$.
- Substitute: $-2\sqrt{28} = -2(2\sqrt{7}) = -4\sqrt{7}$ and $-3\sqrt{112} = -3(4\sqrt{7}) = -12\sqrt{7}$.
- Now the expression is: $-4\sqrt{7} - 12\sqrt{7}$.
- Combine the coefficients: $-4 - 12 = -16$.
- Result: $-16\sqrt{7}$.

$$
\boxed{-16\sqrt{7}}
$$

#### 10) $ -3\sqrt{18} - 3\sqrt{32} $
- Simplify $\sqrt{18}$: $\sqrt{18} = \sqrt{9 \cdot 2} = 3\sqrt{2}$.
- Simplify $\sqrt{32}$: $\sqrt{32} = \sqrt{16 \cdot 2} = 4\sqrt{2}$.
- Substitute: $-3\sqrt{18} = -3(3\sqrt{2}) = -9\sqrt{2}$ and $-3\sqrt{32} = -3(4\sqrt{2}) = -12\sqrt{2}$.
- Now the expression is: $-9\sqrt{2} - 12\sqrt{2}$.
- Combine the coefficients: $-9 - 12 = -21$.
- Result: $-21\sqrt{2}$.

$$
\boxed{-21\sqrt{2}}
$$

#### 11) $ -3\sqrt{3} - \sqrt{3} - 3\sqrt{3} $
- All terms have the same radical part ($\sqrt{3}$).
- Combine the coefficients: $-3 - 1 - 3 = -7$.
- Result: $-7\sqrt{3}$.

$$
\boxed{-7\sqrt{3}}
$$

#### 12) $ -3\sqrt{10} + 5\sqrt{7} - 2\sqrt{10} $
- Combine the $\sqrt{10}$ terms: $-3\sqrt{10} - 2\sqrt{10} = -5\sqrt{10}$.
- The $\sqrt{7}$ term remains as it is.
- Result: $-5\sqrt{10} + 5\sqrt{7}$.

$$
\boxed{-5\sqrt{10} + 5\sqrt{7}}
$$

#### 13) $ -2\sqrt{8} - 5\sqrt{10} - \sqrt{8} $
- Simplify $\sqrt{8}$: $\sqrt{8} = 2\sqrt{2}$.
- Substitute: $-2\sqrt{8} = -2(2\sqrt{2}) = -4\sqrt{2}$ and $-\sqrt{8} = -(2\sqrt{2}) = -2\sqrt{2}$.
- Now the expression is: $-4\sqrt{2} - 5\sqrt{10} - 2\sqrt{2}$.
- Combine the $\sqrt{2}$ terms: $-4\sqrt{2} - 2\sqrt{2} = -6\sqrt{2}$.
- The $\sqrt{10}$ term remains as it is.
- Result: $-6\sqrt{2} - 5\sqrt{10}$.

$$
\boxed{-6\sqrt{2} - 5\sqrt{10}}
$$

#### 14) $ -3\sqrt{6} - 2\sqrt{6} + 3\sqrt{6} $
- All terms have the same radical part ($\sqrt{6}$).
- Combine the coefficients: $-3 - 2 + 3 = -2$.
- Result: $-2\sqrt{6}$.

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

#### 15) $ 3\sqrt{2} - 3\sqrt{7} + 3\sqrt{2} $
- Combine the $\sqrt{2}$ terms: $3\sqrt{2} + 3\sqrt{2} = 6\sqrt{2}$.
- The $\sqrt{7}$ term remains as it is.
- Result: $6\sqrt{2} - 3\sqrt{7}$.

$$
\boxed{6\sqrt{2} - 3\sqrt{7}}
$$

#### 16) $ 2\sqrt{45} - 4\sqrt{63} - 5\sqrt{7} $
- Simplify $\sqrt{45}$: $\sqrt{45} = \sqrt{9 \cdot 5} = 3\sqrt{5}$.
- Simplify $\sqrt{63}$: $\sqrt{63} = \sqrt{9 \cdot 7} = 3\sqrt{7}$.
- Substitute: $2\sqrt{45} = 2(3\sqrt{5}) = 6\sqrt{5}$ and $-4\sqrt{63} = -4(3\sqrt{7}) = -12\sqrt{7}$.
- Now the expression is: $6\sqrt{5} - 12\sqrt{7} - 5\sqrt{7}$.
- Combine the $\sqrt{7}$ terms: $-12\sqrt{7} - 5\sqrt{7} = -17\sqrt{7}$.
- The $\sqrt{5}$ term remains as it is.
- Result: $6\sqrt{5} - 17\sqrt{7}$.

$$
\boxed{6\sqrt{5} - 17\sqrt{7}}
$$

#### 17) $ 5\sqrt{2} - 3\sqrt{18} - 5\sqrt{32} $
- Simplify $\sqrt{18}$: $\sqrt{18} = \sqrt{9 \cdot 2} = 3\sqrt{2}$.
- Simplify $\sqrt{32}$: $\sqrt{32} = \sqrt{16 \cdot 2} = 4\sqrt{2}$.
- Substitute: $-3\sqrt{18} = -3(3\sqrt{2}) = -9\sqrt{2}$ and $-5\sqrt{32} = -5(4\sqrt{2}) = -20\sqrt{2}$.
- Now the expression is: $5\sqrt{2} - 9\sqrt{2} - 20\sqrt{2}$.
- Combine the $\sqrt{2}$ terms: $5 - 9 - 20 = -24$.
- Result: $-24\sqrt{2}$.

$$
\boxed{-24\sqrt{2}}
$$

#### 18) $ 4\sqrt{6} - 3\sqrt{50} - 5\sqrt{18} $
- Simplify $\sqrt{50}$: $\sqrt{50} = \sqrt{25 \cdot 2} = 5\sqrt{2}$.
- Simplify $\sqrt{18}$: $\sqrt{18} = \sqrt{9 \cdot 2} = 3\sqrt{2}$.
- Substitute: $-3\sqrt{50} = -3(5\sqrt{2}) = -15\sqrt{2}$ and $-5\sqrt{18} = -5(3\sqrt{2}) = -15\sqrt{2}$.
- Now the expression is: $4\sqrt{6} - 15\sqrt{2} - 15\sqrt{2}$.
- Combine the $\sqrt{2}$ terms: $-15\sqrt{2} - 15\sqrt{2} = -30\sqrt{2}$.
- The $\sqrt{6}$ term remains as it is.
- Result: $4\sqrt{6} - 30\sqrt{2}$.

$$
\boxed{4\sqrt{6} - 30\sqrt{2}}
$$

#### 19) $ -3\sqrt{40} - 3\sqrt{250} - \sqrt{6} $
- Simplify $\sqrt{40}$: $\sqrt{40} = \sqrt{4 \cdot 10} = 2\sqrt{10}$.
- Simplify $\sqrt{250}$: $\sqrt{250} = \sqrt{25 \cdot 10} = 5\sqrt{10}$.
- Substitute: $-3\sqrt{40} = -3(2\sqrt{10}) = -6\sqrt{10}$ and $-3\sqrt{250} = -3(5\sqrt{10}) = -15\sqrt{10}$.
- Now the expression is: $-6\sqrt{10} - 15\sqrt{10} - \sqrt{6}$.
- Combine the $\sqrt{10}$ terms: $-6 - 15 = -21$.
- The $\sqrt{6}$ term remains as it is.
- Result: $-21\sqrt{10} - \sqrt{6}$.

$$
\boxed{-21\sqrt{10} - \sqrt{6}}
$$

#### 20) $ -4\sqrt{96} - 5\sqrt{6} + 4\sqrt{5} $
- Simplify $\sqrt{96}$: $\sqrt{96} = \sqrt{16 \cdot 6} = 4\sqrt{6}$.
- Substitute: $-4\sqrt{96} = -4(4\sqrt{6}) = -16\sqrt{6}$.
- Now the expression is: $-16\sqrt{6} - 5\sqrt{6} + 4\sqrt{5}$.
- Combine the $\sqrt{6}$ terms: $-16 - 5 = -21$.
- The $\sqrt{5}$ term remains as it is.
- Result: $-21\sqrt{6} + 4\sqrt{5}$.

$$
\boxed{-21\sqrt{6} + 4\sqrt{5}}
$$

#### 21) $ -3\sqrt{10} - 5\sqrt{5} + 5\sqrt{5} $
- Combine the $\sqrt{5}$ terms: $-5\sqrt{5} + 5\sqrt{5} = 0$.
- The $\sqrt{10}$ term remains as it is.
- Result: $-3\sqrt{10}$.

$$
\boxed{-3\sqrt{10}}
$$

#### 22) $ -3\sqrt{50} - 2\sqrt{2} + 4\sqrt{7} $
- Simplify $\sqrt{50}$: $\sqrt{50} = \sqrt{25 \cdot 2} = 5\sqrt{2}$.
- Substitute: $-3\sqrt{50} = -3(5\sqrt{2}) = -15\sqrt{2}$.
- Now the expression is: $-15\sqrt{2} - 2\sqrt{2} + 4\sqrt{7}$.
- Combine the $\sqrt{2}$ terms: $-15 - 2 = -17$.
- The $\sqrt{7}$ term remains as it is.
- Result: $-17\sqrt{2} + 4\sqrt{7}$.

$$
\boxed{-17\sqrt{2} + 4\sqrt{7}}
$$

#### 23) $ 2\sqrt{5} + 5\sqrt{32} - 5\sqrt{200} $
- Simplify $\sqrt{32}$: $\sqrt{32} = \sqrt{16 \cdot 2} = 4\sqrt{2}$.
- Simplify $\sqrt{200}$: $\sqrt{200} = \sqrt{100 \cdot 2} = 10\sqrt{2}$.
- Substitute: $5\sqrt{32} = 5(4\sqrt{2}) = 20\sqrt{2}$ and $-5\sqrt{200} = -5(10\sqrt{2}) = -50\sqrt{2}$.
- Now the expression is: $2\sqrt{5} + 20\sqrt{2} - 50\sqrt{2}$.
- Combine the $\sqrt{2}$ terms: $20 - 50 = -30$.
- The $\sqrt{5}$ term remains as it is.
- Result: $2\sqrt{5} - 30\sqrt{2}$.

$$
\boxed{2\sqrt{5} - 30\sqrt{2}}
$$

#### 24) $ -\sqrt{27} + 4\sqrt{24} + 5\sqrt{27} $
- Simplify $\sqrt{27}$: $\sqrt{27} = \sqrt{9 \cdot 3} = 3\sqrt{3}$.
- Simplify $\sqrt{24}$: $\sqrt{24} = \sqrt{4 \cdot 6} = 2\sqrt{6}$.
- Substitute: $-\sqrt{27} = -(3\sqrt{3}) = -3\sqrt{3}$, $4\sqrt{24} = 4(2\sqrt{6}) = 8\sqrt{6}$, and $5\sqrt{27} = 5(3\sqrt{3}) = 15\sqrt{3}$.
- Now the expression is: $-3\sqrt{3} + 8\sqrt{6} + 15\sqrt{3}$.
- Combine the $\sqrt{3}$ terms: $-3 + 15 = 12$.
- The $\sqrt{6}$ term remains as it is.
- Result: $12\sqrt{3} + 8\sqrt{6}$.

$$
\boxed{12\sqrt{3} + 8\sqrt{6}}
$$

#### 25) $ 4\sqrt{24} - 5\sqrt{160} - 2\sqrt{24} $
- Simplify $\sqrt{24}$: $\sqrt{24} = \sqrt{4 \cdot 6} = 2\sqrt{6}$.
- Simplify $\sqrt{160}$: $\sqrt{160} = \sqrt{16 \cdot 10} = 4\sqrt{10}$.
- Substitute: $4\sqrt{24} = 4(2\sqrt{6}) = 8\sqrt{6}$, $-5\sqrt{160} = -5(4\sqrt{10}) = -20\sqrt{10}$, and $-2\sqrt{24} = -2(2\sqrt{6}) = -4\sqrt{6}$.
- Now the expression is: $8\sqrt{6} - 20\sqrt{10} - 4\sqrt{6}$.
- Combine the $\sqrt{6}$ terms: $8 - 4 = 4$.
- The $\sqrt{10}$ term remains as it is.
- Result: $4\sqrt{6} - 20\sqrt{10}$.

$$
\boxed{4\sqrt{6} - 20\sqrt{10}}
$$

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Final Answers:

1. $\boxed{-\sqrt{6}}$
2. $\boxed{-3\sqrt{7}}$
3. $\boxed{-7\sqrt{2}}$
4. $\boxed{0}$
5. $\boxed{-8\sqrt{6}}$
6. $\boxed{\sqrt{10}}$
7. $\boxed{-42\sqrt{2}}$
8. $\boxed{-2\sqrt{5}}$
9. $\boxed{-16\sqrt{7}}$
10. $\boxed{-21\sqrt{2}}$
11. $\boxed{-7\sqrt{3}}$
12. $\boxed{-5\sqrt{10} + 5\sqrt{7}}$
13. $\boxed{-6\sqrt{2} - 5\sqrt{10}}$
14. $\boxed{-2\sqrt{6}}$
15. $\boxed{6\sqrt{2} - 3\sqrt{7}}$
16. $\boxed{6\sqrt{5} - 17\sqrt{7}}$
17. $\boxed{-24\sqrt{2}}$
18. $\boxed{4\sqrt{6} - 30\sqrt{2}}$
19. $\boxed{-21\sqrt{10} - \sqrt{6}}$
20. $\boxed{-21\sqrt{6} + 4\sqrt{5}}$
21. $\boxed{-3\sqrt{10}}$
22. $\boxed{-17\sqrt{2} + 4\sqrt{7}}$
23. $\boxed{2\sqrt{5} - 30\sqrt{2}}$
24. $\boxed{12\sqrt{3} + 8\sqrt{6}}$
25. $\boxed{4\sqrt{6} - 20\sqrt{10}}$