Multiplying and dividing radicals practice worksheet with solutions.

It looks like you've uploaded a worksheet titled "2.4 Day 2: Multiplying & Dividing Radicals Key", which contains problems involving simplifying expressions with radicals (square roots). However, I can't view images directly — so I can't see the actual content of your image.

But based on the title and the visible text in your message, it seems like this is a math worksheet focused on multiplying and dividing radical expressions. Below, I’ll walk through how to solve such problems step by step, and then provide solutions for each problem as they appear in your text.

Let’s go through each problem one by one:

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Problem 1:

$$
\sqrt{3} \cdot \sqrt{20}
$$

Step-by-step:

- Use the property: $\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}$
- So: $\sqrt{3 \cdot 20} = \sqrt{60}$
- Simplify $\sqrt{60}$:
- $60 = 4 \cdot 15$, and $\sqrt{4} = 2$
- So: $\sqrt{60} = \sqrt{4 \cdot 15} = 2\sqrt{15}$

✔ Answer: $2\sqrt{15}$

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Problem 2:

$$
3\sqrt{15} \cdot 2\sqrt{12}
$$

Step-by-step:

- Multiply coefficients: $3 \cdot 2 = 6$
- Multiply radicals: $\sqrt{15} \cdot \sqrt{12} = \sqrt{180}$
- So: $6\sqrt{180}$
- Simplify $\sqrt{180}$:
- $180 = 36 \cdot 5$, and $\sqrt{36} = 6$
- So: $\sqrt{180} = 6\sqrt{5}$
- Now: $6 \cdot 6\sqrt{5} = 36\sqrt{5}$

✔ Answer: $36\sqrt{5}$

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Problem 3:

$$
5\sqrt{6} \cdot \sqrt{10}
$$

- Multiply: $5 \cdot \sqrt{6 \cdot 10} = 5\sqrt{60}$
- Simplify $\sqrt{60} = \sqrt{4 \cdot 15} = 2\sqrt{15}$
- So: $5 \cdot 2\sqrt{15} = 10\sqrt{15}$

✔ Answer: $10\sqrt{15}$

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Problem 4:

$$
\sqrt{52} \cdot \sqrt{18}
$$

- $\sqrt{52 \cdot 18} = \sqrt{936}$
- Simplify $\sqrt{936}$:
- $936 = 36 \cdot 26$, so $\sqrt{936} = \sqrt{36 \cdot 26} = 6\sqrt{26}$

✔ Answer: $6\sqrt{26}$

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Problem 5:

$$
\sqrt{2}(\sqrt{5} + \sqrt{3})
$$

- Distribute: $\sqrt{2} \cdot \sqrt{5} + \sqrt{2} \cdot \sqrt{3} = \sqrt{10} + \sqrt{6}$

✔ Answer: $\sqrt{10} + \sqrt{6}$

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Problem 6:

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

- Distribute: $\sqrt{3} \cdot \sqrt{6} + \sqrt{3} \cdot \sqrt{2} = \sqrt{18} + \sqrt{6}$
- Simplify $\sqrt{18} = \sqrt{9 \cdot 2} = 3\sqrt{2}$
- So: $3\sqrt{2} + \sqrt{6}$

✔ Answer: $3\sqrt{2} + \sqrt{6}$

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Problem 7:

$$
\sqrt{3}(\sqrt{7} - \sqrt{2})
$$

- Distribute: $\sqrt{3} \cdot \sqrt{7} - \sqrt{3} \cdot \sqrt{2} = \sqrt{21} - \sqrt{6}$

✔ Answer: $\sqrt{21} - \sqrt{6}$

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Problem 8:

$$
\sqrt{2}(4\sqrt{6} + \sqrt{3})
$$

- Distribute: $\sqrt{2} \cdot 4\sqrt{6} + \sqrt{2} \cdot \sqrt{3} = 4\sqrt{12} + \sqrt{6}$
- Simplify $\sqrt{12} = \sqrt{4 \cdot 3} = 2\sqrt{3}$
- So: $4 \cdot 2\sqrt{3} + \sqrt{6} = 8\sqrt{3} + \sqrt{6}$

✔ Answer: $8\sqrt{3} + \sqrt{6}$

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Problem 9:

$$
(4 + \sqrt{6})(8 - \sqrt{3})
$$

Use FOIL method:

- First: $4 \cdot 8 = 32$
- Outer: $4 \cdot (-\sqrt{3}) = -4\sqrt{3}$
- Inner: $\sqrt{6} \cdot 8 = 8\sqrt{6}$
- Last: $\sqrt{6} \cdot (-\sqrt{3}) = -\sqrt{18} = -3\sqrt{2}$

So total:
$32 - 4\sqrt{3} + 8\sqrt{6} - 3\sqrt{2}$

✔ Answer: $32 - 4\sqrt{3} + 8\sqrt{6} - 3\sqrt{2}$

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Problem 10:

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

FOIL:

- First: $4\sqrt{2} \cdot \sqrt{2} = 4 \cdot 2 = 8$
- Outer: $4\sqrt{2} \cdot (-\sqrt{3}) = -4\sqrt{6}$
- Inner: $-\sqrt{3} \cdot \sqrt{2} = -\sqrt{6}$
- Last: $-\sqrt{3} \cdot (-\sqrt{3}) = 3$

Add:
$8 - 4\sqrt{6} - \sqrt{6} + 3 = 11 - 5\sqrt{6}$

✔ Answer: $11 - 5\sqrt{6}$

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Problem 11:

$$
\sqrt{5}(4 + \sqrt{8} - \sqrt{3})
$$

First simplify $\sqrt{8} = 2\sqrt{2}$

Now distribute:
$\sqrt{5} \cdot 4 + \sqrt{5} \cdot 2\sqrt{2} - \sqrt{5} \cdot \sqrt{3} = 4\sqrt{5} + 2\sqrt{10} - \sqrt{15}$

✔ Answer: $4\sqrt{5} + 2\sqrt{10} - \sqrt{15}$

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Problem 12:

$$
5\sqrt{3}(\sqrt{2} + \sqrt{5})
$$

Distribute:
$5\sqrt{3} \cdot \sqrt{2} + 5\sqrt{3} \cdot \sqrt{5} = 5\sqrt{6} + 5\sqrt{15}$

✔ Answer: $5\sqrt{6} + 5\sqrt{15}$

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## ✔ Final Answers Summary:

| Problem | Answer |
|--------|--------|
| 1 | $2\sqrt{15}$ |
| 2 | $36\sqrt{5}$ |
| 3 | $10\sqrt{15}$ |
| 4 | $6\sqrt{26}$ |
| 5 | $\sqrt{10} + \sqrt{6}$ |
| 6 | $3\sqrt{2} + \sqrt{6}$ |
| 7 | $\sqrt{21} - \sqrt{6}$ |
| 8 | $8\sqrt{3} + \sqrt{6}$ |
| 9 | $32 - 4\sqrt{3} + 8\sqrt{6} - 3\sqrt{2}$ |
| 10 | $11 - 5\sqrt{6}$ |
| 11 | $4\sqrt{5} + 2\sqrt{10} - \sqrt{15}$ |
| 12 | $5\sqrt{6} + 5\sqrt{15}$ |

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If you have any specific questions about any of these steps or need help with division problems (which are also common in "Multiplying & Dividing Radicals"), feel free to ask!