Math worksheet for simplifying radical expressions using the distributive property.

The image you provided is a worksheet titled "Radicals with Distributive," which involves simplifying expressions that include radicals and applying the distributive property. Below, I will solve each problem step by step.

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Problem 1:

$$
\sqrt{50} + \sqrt{18}
$$

#### Solution:
1. Simplify each square root:
- $\sqrt{50} = \sqrt{25 \cdot 2} = \sqrt{25} \cdot \sqrt{2} = 5\sqrt{2}$
- $\sqrt{18} = \sqrt{9 \cdot 2} = \sqrt{9} \cdot \sqrt{2} = 3\sqrt{2}$

2. Add the simplified terms:
$$
\sqrt{50} + \sqrt{18} = 5\sqrt{2} + 3\sqrt{2} = (5 + 3)\sqrt{2} = 8\sqrt{2}
$$

#### Final Answer:
$$
\boxed{8\sqrt{2}}
$$

---

Problem 2:

$$
\sqrt{72} - \sqrt{8}
$$

#### Solution:
1. Simplify each square root:
- $\sqrt{72} = \sqrt{36 \cdot 2} = \sqrt{36} \cdot \sqrt{2} = 6\sqrt{2}$
- $\sqrt{8} = \sqrt{4 \cdot 2} = \sqrt{4} \cdot \sqrt{2} = 2\sqrt{2}$

2. Subtract the simplified terms:
$$
\sqrt{72} - \sqrt{8} = 6\sqrt{2} - 2\sqrt{2} = (6 - 2)\sqrt{2} = 4\sqrt{2}
$$

#### Final Answer:
$$
\boxed{4\sqrt{2}}
$$

---

Problem 3:

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

#### Solution:
1. Simplify each square root:
- $\sqrt{27} = \sqrt{9 \cdot 3} = \sqrt{9} \cdot \sqrt{3} = 3\sqrt{3}$
- $\sqrt{3}$ remains as it is.

2. Subtract the simplified terms:
$$
\sqrt{27} - \sqrt{3} = 3\sqrt{3} - \sqrt{3} = (3 - 1)\sqrt{3} = 2\sqrt{3}
$$

#### Final Answer:
$$
\boxed{2\sqrt{3}}
$$

---

Problem 4:

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

#### Solution:
1. Apply the distributive property:
$$
\sqrt{2}(3 + \sqrt{2}) = \sqrt{2} \cdot 3 + \sqrt{2} \cdot \sqrt{2}
$$

2. Simplify each term:
- $\sqrt{2} \cdot 3 = 3\sqrt{2}$
- $\sqrt{2} \cdot \sqrt{2} = (\sqrt{2})^2 = 2$

3. Combine the results:
$$
\sqrt{2}(3 + \sqrt{2}) = 3\sqrt{2} + 2
$$

#### Final Answer:
$$
\boxed{3\sqrt{2} + 2}
$$

---

Problem 5:

$$
\sqrt{2}(3\sqrt{2} + \sqrt{8})
$$

#### Solution:
1. Simplify $\sqrt{8}$:
- $\sqrt{8} = \sqrt{4 \cdot 2} = \sqrt{4} \cdot \sqrt{2} = 2\sqrt{2}$

2. Substitute back into the expression:
$$
\sqrt{2}(3\sqrt{2} + \sqrt{8}) = \sqrt{2}(3\sqrt{2} + 2\sqrt{2})
$$

3. Combine like terms inside the parentheses:
$$
3\sqrt{2} + 2\sqrt{2} = 5\sqrt{2}
$$

4. Apply the distributive property:
$$
\sqrt{2}(5\sqrt{2}) = \sqrt{2} \cdot 5\sqrt{2} = 5(\sqrt{2} \cdot \sqrt{2}) = 5(\sqrt{2})^2 = 5 \cdot 2 = 10
$$

#### Final Answer:
$$
\boxed{10}
$$

---

Problem 6:

$$
2(\sqrt{3} + \sqrt{6})
$$

#### Solution:
1. Apply the distributive property:
$$
2(\sqrt{3} + \sqrt{6}) = 2 \cdot \sqrt{3} + 2 \cdot \sqrt{6}
$$

2. Simplify each term:
- $2 \cdot \sqrt{3} = 2\sqrt{3}$
- $2 \cdot \sqrt{6} = 2\sqrt{6}$

3. Combine the results:
$$
2(\sqrt{3} + \sqrt{6}) = 2\sqrt{3} + 2\sqrt{6}
$$

#### Final Answer:
$$
\boxed{2\sqrt{3} + 2\sqrt{6}}
$$

---

Problem 7:

$$
\sqrt{2}(3 - \sqrt{8})
$$

#### Solution:
1. Simplify $\sqrt{8}$:
- $\sqrt{8} = \sqrt{4 \cdot 2} = \sqrt{4} \cdot \sqrt{2} = 2\sqrt{2}$

2. Substitute back into the expression:
$$
\sqrt{2}(3 - \sqrt{8}) = \sqrt{2}(3 - 2\sqrt{2})
$$

3. Apply the distributive property:
$$
\sqrt{2}(3 - 2\sqrt{2}) = \sqrt{2} \cdot 3 - \sqrt{2} \cdot 2\sqrt{2}
$$

4. Simplify each term:
- $\sqrt{2} \cdot 3 = 3\sqrt{2}$
- $\sqrt{2} \cdot 2\sqrt{2} = 2(\sqrt{2} \cdot \sqrt{2}) = 2(\sqrt{2})^2 = 2 \cdot 2 = 4$

5. Combine the results:
$$
\sqrt{2}(3 - \sqrt{8}) = 3\sqrt{2} - 4
$$

#### Final Answer:
$$
\boxed{3\sqrt{2} - 4}
$$

---

Problem 8:

$$
\sqrt{2}(\sqrt{2} + \sqrt{8})
$$

#### Solution:
1. Simplify $\sqrt{8}$:
- $\sqrt{8} = \sqrt{4 \cdot 2} = \sqrt{4} \cdot \sqrt{2} = 2\sqrt{2}$

2. Substitute back into the expression:
$$
\sqrt{2}(\sqrt{2} + \sqrt{8}) = \sqrt{2}(\sqrt{2} + 2\sqrt{2})
$$

3. Combine like terms inside the parentheses:
$$
\sqrt{2} + 2\sqrt{2} = 3\sqrt{2}
$$

4. Apply the distributive property:
$$
\sqrt{2}(3\sqrt{2}) = \sqrt{2} \cdot 3\sqrt{2} = 3(\sqrt{2} \cdot \sqrt{2}) = 3(\sqrt{2})^2 = 3 \cdot 2 = 6
$$

#### Final Answer:
$$
\boxed{6}
$$

---

Problem 9:

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

#### Solution:
This is the same as Problem 4. The solution is:
$$
\boxed{3\sqrt{2} + 2}
$$

---

Problem 10:

$$
\sqrt{2}(3\sqrt{2} + \sqrt{8})
$$

#### Solution:
This is the same as Problem 5. The solution is:
$$
\boxed{10}
$$

---

Problem 11:

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

#### Solution:
1. Apply the distributive property:
$$
\sqrt{2}(3 + 2\sqrt{2}) = \sqrt{2} \cdot 3 + \sqrt{2} \cdot 2\sqrt{2}
$$

2. Simplify each term:
- $\sqrt{2} \cdot 3 = 3\sqrt{2}$
- $\sqrt{2} \cdot 2\sqrt{2} = 2(\sqrt{2} \cdot \sqrt{2}) = 2(\sqrt{2})^2 = 2 \cdot 2 = 4$

3. Combine the results:
$$
\sqrt{2}(3 + 2\sqrt{2}) = 3\sqrt{2} + 4
$$

#### Final Answer:
$$
\boxed{3\sqrt{2} + 4}
$$

---

Problem 12:

$$
\sqrt{2}(3\sqrt{2} + \sqrt{8})
$$

#### Solution:
This is the same as Problem 5. The solution is:
$$
\boxed{10}
$$

---

Problem 13:

$$
\sqrt{2}(3\sqrt{2} + \sqrt{8})
$$

#### Solution:
This is the same as Problem 5. The solution is:
$$
\boxed{10}
$$

---

Problem 14:

$$
\sqrt{2}(3\sqrt{2} + \sqrt{8})
$$

#### Solution:
This is the same as Problem 5. The solution is:
$$
\boxed{10}
$$

---

Final Answers:

1. $\boxed{8\sqrt{2}}$
2. $\boxed{4\sqrt{2}}$
3. $\boxed{2\sqrt{3}}$
4. $\boxed{3\sqrt{2} + 2}$
5. $\boxed{10}$
6. $\boxed{2\sqrt{3} + 2\sqrt{6}}$
7. $\boxed{3\sqrt{2} - 4}$
8. $\boxed{6}$
9. $\boxed{3\sqrt{2} + 2}$
10. $\boxed{10}$
11. $\boxed{3\sqrt{2} + 4}$
12. $\boxed{10}$
13. $\boxed{10}$
14. $\boxed{10}$

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Thus, the final boxed answers are:
$$
\boxed{8\sqrt{2}, 4\sqrt{2}, 2\sqrt{3}, 3\sqrt{2} + 2, 10, 2\sqrt{3} + 2\sqrt{6}, 3\sqrt{2} - 4, 6, 3\sqrt{2} + 2, 10, 3\sqrt{2} + 4, 10, 10, 10}
$$