Practice worksheet for adding and subtracting radical expressions with simplification problems.

To solve the problems involving adding, subtracting, and multiplying radical expressions, we need to follow specific rules. Let's go through each type of problem step by step.

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1. Adding or Subtracting Radical Expressions

#### Rule:
- You can only add or subtract radical expressions if they have the same radicand (the number under the square root). If the radicands are different, you cannot combine them.

#### Examples:

1. Simplify: \( 4\sqrt{5} + 3\sqrt{5} \)
- Both terms have the same radicand (\(\sqrt{5}\)).
- Combine the coefficients: \( 4 + 3 = 7 \).
- Result: \( 7\sqrt{5} \).

2. Simplify: \( 6\sqrt{3} + 2\sqrt{3} \)
- Both terms have the same radicand (\(\sqrt{3}\)).
- Combine the coefficients: \( 6 + 2 = 8 \).
- Result: \( 8\sqrt{3} \).

3. Simplify: \( 16\sqrt{2} + 9\sqrt{2} \)
- Both terms have the same radicand (\(\sqrt{2}\)).
- Combine the coefficients: \( 16 + 9 = 25 \).
- Result: \( 25\sqrt{2} \).

4. Simplify: \( 7\sqrt{3} + 2\sqrt{3} \)
- Both terms have the same radicand (\(\sqrt{3}\)).
- Combine the coefficients: \( 7 + 2 = 9 \).
- Result: \( 9\sqrt{3} \).

5. Simplify: \( 4\sqrt{7} + \sqrt{7} \)
- Both terms have the same radicand (\(\sqrt{7}\)).
- Combine the coefficients: \( 4 + 1 = 5 \).
- Result: \( 5\sqrt{7} \).

6. Simplify: \( 3\sqrt{11} + 4\sqrt{11} \)
- Both terms have the same radicand (\(\sqrt{11}\)).
- Combine the coefficients: \( 3 + 4 = 7 \).
- Result: \( 7\sqrt{11} \).

7. Simplify: \( 5\sqrt{13} + 3\sqrt{13} \)
- Both terms have the same radicand (\(\sqrt{13}\)).
- Combine the coefficients: \( 5 + 3 = 8 \).
- Result: \( 8\sqrt{13} \).

8. Simplify: \( 6\sqrt{4} + 4\sqrt{4} \)
- Both terms have the same radicand (\(\sqrt{4}\)).
- Simplify \(\sqrt{4}\): \(\sqrt{4} = 2\).
- Combine the coefficients: \( 6 + 4 = 10 \).
- Result: \( 10 \cdot 2 = 20 \).

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2. Multiplying Radical Expressions

#### Rule:
- When multiplying radical expressions, you can multiply the coefficients together and the radicands together.
- Use the property: \( \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \).

#### Examples:

1. Simplify: \( (3\sqrt{2})(2\sqrt{3}) \)
- Multiply the coefficients: \( 3 \cdot 2 = 6 \).
- Multiply the radicands: \( \sqrt{2} \cdot \sqrt{3} = \sqrt{6} \).
- Result: \( 6\sqrt{6} \).

2. Simplify: \( (2\sqrt{5})(3\sqrt{2}) \)
- Multiply the coefficients: \( 2 \cdot 3 = 6 \).
- Multiply the radicands: \( \sqrt{5} \cdot \sqrt{2} = \sqrt{10} \).
- Result: \( 6\sqrt{10} \).

3. Simplify: \( (4\sqrt{3})(5\sqrt{7}) \)
- Multiply the coefficients: \( 4 \cdot 5 = 20 \).
- Multiply the radicands: \( \sqrt{3} \cdot \sqrt{7} = \sqrt{21} \).
- Result: \( 20\sqrt{21} \).

4. Simplify: \( (5\sqrt{3})^2 \)
- Square both the coefficient and the radicand:
\[
(5\sqrt{3})^2 = 5^2 \cdot (\sqrt{3})^2 = 25 \cdot 3 = 75.
\]
- Result: \( 75 \).

5. Simplify: \( (-\sqrt{5})(x + 2\sqrt{5}) \)
- Distribute \(-\sqrt{5}\):
\[
(-\sqrt{5})(x) + (-\sqrt{5})(2\sqrt{5}) = -x\sqrt{5} - 2(\sqrt{5} \cdot \sqrt{5}) = -x\sqrt{5} - 2 \cdot 5 = -x\sqrt{5} - 10.
\]
- Result: \( -x\sqrt{5} - 10 \).

6. Simplify: \( (2\sqrt{3})(x - \sqrt{3}) \)
- Distribute \(2\sqrt{3}\):
\[
(2\sqrt{3})(x) + (2\sqrt{3})(-\sqrt{3}) = 2x\sqrt{3} - 2(\sqrt{3} \cdot \sqrt{3}) = 2x\sqrt{3} - 2 \cdot 3 = 2x\sqrt{3} - 6.
\]
- Result: \( 2x\sqrt{3} - 6 \).

7. Simplify: \( (3\sqrt{2})(x + 3\sqrt{2}) \)
- Distribute \(3\sqrt{2}\):
\[
(3\sqrt{2})(x) + (3\sqrt{2})(3\sqrt{2}) = 3x\sqrt{2} + 3 \cdot 3 \cdot (\sqrt{2} \cdot \sqrt{2}) = 3x\sqrt{2} + 9 \cdot 2 = 3x\sqrt{2} + 18.
\]
- Result: \( 3x\sqrt{2} + 18 \).

8. Simplify: \( (2 - 3\sqrt{5})(2 + 3\sqrt{5}) \)
- Use the difference of squares formula: \( (a - b)(a + b) = a^2 - b^2 \).
\[
(2 - 3\sqrt{5})(2 + 3\sqrt{5}) = 2^2 - (3\sqrt{5})^2 = 4 - 9 \cdot 5 = 4 - 45 = -41.
\]
- Result: \( -41 \).

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Final Answers:

1. \( 7\sqrt{5} \)
2. \( 8\sqrt{3} \)
3. \( 25\sqrt{2} \)
4. \( 9\sqrt{3} \)
5. \( 5\sqrt{7} \)
6. \( 7\sqrt{11} \)
7. \( 8\sqrt{13} \)
8. \( 20 \)

9. \( 6\sqrt{6} \)
10. \( 6\sqrt{10} \)
11. \( 20\sqrt{21} \)
12. \( 75 \)
13. \( -x\sqrt{5} - 10 \)
14. \( 2x\sqrt{3} - 6 \)
15. \( 3x\sqrt{2} + 18 \)
16. \( -41 \)

Boxed Final Answer:
\[
\boxed{
\begin{array}{ll}
1. & 7\sqrt{5} \\
2. & 8\sqrt{3} \\
3. & 25\sqrt{2} \\
4. & 9\sqrt{3} \\
5. & 5\sqrt{7} \\
6. & 7\sqrt{11} \\
7. & 8\sqrt{13} \\
8. & 20 \\
9. & 6\sqrt{6} \\
10. & 6\sqrt{10} \\
11. & 20\sqrt{21} \\
12. & 75 \\
13. & -x\sqrt{5} - 10 \\
14. & 2x\sqrt{3} - 6 \\
15. & 3x\sqrt{2} + 18 \\
16. & -41 \\
\end{array}
}
\]