Math worksheet for practicing multiplication and division of radical expressions, featuring 15 problems to simplify.

Let's solve each problem step by step, starting with the basic multiplication and division of radical expressions, followed by simplifying the more complex expressions.

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Part 1: Multiply and Divide Radical Expressions

We'll evaluate these basic operations:

1. $ 2 \cdot 5 = \boxed{10} $

2. $ 2 \cdot \sqrt{5} = \boxed{2\sqrt{5}} $

3. $ \sqrt{2} \cdot 5 = \boxed{5\sqrt{2}} $

4. $ \sqrt{2} \cdot \sqrt{5} = \sqrt{2 \cdot 5} = \boxed{\sqrt{10}} $

5. $ 2\sqrt{3} \cdot 5 = 10\sqrt{3} $ → $ \boxed{10\sqrt{3}} $

6. $ 2\sqrt{3} \cdot \sqrt{6} = 2 \cdot \sqrt{3 \cdot 6} = 2\sqrt{18} = 2 \cdot \sqrt{9 \cdot 2} = 2 \cdot 3\sqrt{2} = \boxed{6\sqrt{2}} $

7. $ 2\sqrt{3} \cdot 4\sqrt{5} = (2 \cdot 4) \cdot \sqrt{3 \cdot 5} = 8\sqrt{15} $ → $ \boxed{8\sqrt{15}} $

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Part 2: Perform the indicated operations and simplify

We now simplify each expression:

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1. $ \sqrt{5} \cdot \sqrt{7} $
$$
= \sqrt{5 \cdot 7} = \boxed{\sqrt{35}}
$$

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2. $ \sqrt{3} \cdot \sqrt{21} $
$$
= \sqrt{3 \cdot 21} = \sqrt{63} = \sqrt{9 \cdot 7} = 3\sqrt{7} \rightarrow \boxed{3\sqrt{7}}
$$

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3. $ \sqrt{10} \cdot \sqrt{50} $
$$
= \sqrt{10 \cdot 50} = \sqrt{500} = \sqrt{100 \cdot 5} = 10\sqrt{5} \rightarrow \boxed{10\sqrt{5}}
$$

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4. $ 4(\sqrt{2} - \sqrt{7}) $
Distribute:
$$
= 4\sqrt{2} - 4\sqrt{7} \rightarrow \boxed{4\sqrt{2} - 4\sqrt{7}}
$$

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5. $ \sqrt{5}(6 - \sqrt{5}) $
Distribute:
$$
= 6\sqrt{5} - \sqrt{5} \cdot \sqrt{5} = 6\sqrt{5} - 5 \rightarrow \boxed{6\sqrt{5} - 5}
$$

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6. $ 2\sqrt{3}(2\sqrt{3} - 4\sqrt{5}) $
Distribute:
$$
= 2\sqrt{3} \cdot 2\sqrt{3} - 2\sqrt{3} \cdot 4\sqrt{5} = 4 \cdot 3 - 8\sqrt{15} = 12 - 8\sqrt{15} \rightarrow \boxed{12 - 8\sqrt{15}}
$$

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7. $ \sqrt{7}(4\sqrt{7} - 2\sqrt{3}) $
Distribute:
$$
= \sqrt{7} \cdot 4\sqrt{7} - \sqrt{7} \cdot 2\sqrt{3} = 4 \cdot 7 - 2\sqrt{21} = 28 - 2\sqrt{21} \rightarrow \boxed{28 - 2\sqrt{21}}
$$

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8. $ \sqrt{3x}(\sqrt{6x} - \sqrt{12}) $
Distribute:
$$
= \sqrt{3x} \cdot \sqrt{6x} - \sqrt{3x} \cdot \sqrt{12}
$$

First term:
$$
\sqrt{3x \cdot 6x} = \sqrt{18x^2} = \sqrt{9 \cdot 2 \cdot x^2} = 3x\sqrt{2}
$$

Second term:
$$
\sqrt{3x \cdot 12} = \sqrt{36x} = \sqrt{36} \cdot \sqrt{x} = 6\sqrt{x}
$$

So:
$$
= 3x\sqrt{2} - 6\sqrt{x} \rightarrow \boxed{3x\sqrt{2} - 6\sqrt{x}}
$$

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9. $ 3\sqrt{2}(\sqrt{2} - 4) + \sqrt{2}(5 - \sqrt{2}) $
Break into two parts:

First part:
$$
3\sqrt{2} \cdot \sqrt{2} - 3\sqrt{2} \cdot 4 = 3 \cdot 2 - 12\sqrt{2} = 6 - 12\sqrt{2}
$$

Second part:
$$
\sqrt{2} \cdot 5 - \sqrt{2} \cdot \sqrt{2} = 5\sqrt{2} - 2
$$

Now add both:
$$
(6 - 12\sqrt{2}) + (5\sqrt{2} - 2) = (6 - 2) + (-12\sqrt{2} + 5\sqrt{2}) = 4 - 7\sqrt{2} \rightarrow \boxed{4 - 7\sqrt{2}}
$$

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10. $ (\sqrt{6} - 3)(\sqrt{6} + 4) $
Use distributive property (FOIL):

$$
= \sqrt{6} \cdot \sqrt{6} + \sqrt{6} \cdot 4 - 3 \cdot \sqrt{6} - 3 \cdot 4
= 6 + 4\sqrt{6} - 3\sqrt{6} - 12
= (6 - 12) + (4\sqrt{6} - 3\sqrt{6}) = -6 + \sqrt{6} \rightarrow \boxed{-6 + \sqrt{6}}
$$

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11. $ (\sqrt{m} - \sqrt{5})^2 $
Use formula: $ (a - b)^2 = a^2 - 2ab + b^2 $

$$
= (\sqrt{m})^2 - 2\sqrt{m}\sqrt{5} + (\sqrt{5})^2 = m - 2\sqrt{5m} + 5 \rightarrow \boxed{m + 5 - 2\sqrt{5m}}
$$

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12. $ (5\sqrt{x} + 2)(2\sqrt{x} - 1) $
FOIL:

$$
= 5\sqrt{x} \cdot 2\sqrt{x} + 5\sqrt{x} \cdot (-1) + 2 \cdot 2\sqrt{x} + 2 \cdot (-1)
= 10x - 5\sqrt{x} + 4\sqrt{x} - 2
= 10x - \sqrt{x} - 2 \rightarrow \boxed{10x - \sqrt{x} - 2}
$$

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13. $ (\sqrt{5} - x)(\sqrt{5} + x) $
This is a difference of squares:

$$
= (\sqrt{5})^2 - x^2 = 5 - x^2 \rightarrow \boxed{5 - x^2}
$$

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14. $ (5\sqrt{2} + 3)(\sqrt{2} - 3) $
FOIL:

$$
= 5\sqrt{2} \cdot \sqrt{2} + 5\sqrt{2} \cdot (-3) + 3 \cdot \sqrt{2} + 3 \cdot (-3)
= 5 \cdot 2 - 15\sqrt{2} + 3\sqrt{2} - 9
= 10 - 15\sqrt{2} + 3\sqrt{2} - 9
= (10 - 9) + (-15\sqrt{2} + 3\sqrt{2}) = 1 - 12\sqrt{2} \rightarrow \boxed{1 - 12\sqrt{2}}
$$

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15. $ (3 + 2\sqrt{5})^2 $
Use $ (a + b)^2 = a^2 + 2ab + b^2 $

$$
= 3^2 + 2 \cdot 3 \cdot 2\sqrt{5} + (2\sqrt{5})^2
= 9 + 12\sqrt{5} + 4 \cdot 5
= 9 + 12\sqrt{5} + 20 = 29 + 12\sqrt{5} \rightarrow \boxed{29 + 12\sqrt{5}}
$$

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✔ Final Answers Summary:

#### Part 1:
1. $ 10 $
2. $ 2\sqrt{5} $
3. $ 5\sqrt{2} $
4. $ \sqrt{10} $
5. $ 10\sqrt{3} $
6. $ 6\sqrt{2} $
7. $ 8\sqrt{15} $

#### Part 2:
1. $ \sqrt{35} $
2. $ 3\sqrt{7} $
3. $ 10\sqrt{5} $
4. $ 4\sqrt{2} - 4\sqrt{7} $
5. $ 6\sqrt{5} - 5 $
6. $ 12 - 8\sqrt{15} $
7. $ 28 - 2\sqrt{21} $
8. $ 3x\sqrt{2} - 6\sqrt{x} $
9. $ 4 - 7\sqrt{2} $
10. $ -6 + \sqrt{6} $
11. $ m + 5 - 2\sqrt{5m} $
12. $ 10x - \sqrt{x} - 2 $
13. $ 5 - x^2 $
14. $ 1 - 12\sqrt{2} $
15. $ 29 + 12\sqrt{5} $

Let me know if you'd like a visual explanation or further breakdown!