Math worksheet for simplifying radical expressions with a color-by-number activity.

To solve the given problems involving radical expressions, we will simplify each expression step by step. Here are the solutions:

---

Problem 1: \( 6\sqrt{3} - 4\sqrt{3} - 10\sqrt{3} \)

- Combine like terms:
\[
6\sqrt{3} - 4\sqrt{3} - 10\sqrt{3} = (6 - 4 - 10)\sqrt{3} = -8\sqrt{3}
\]
- Simplified Expression: \(-8\sqrt{3}\)
- Color: Lilac

---

Problem 2: \( \sqrt{60} + 5\sqrt{60} \)

- Combine like terms:
\[
\sqrt{60} + 5\sqrt{60} = (1 + 5)\sqrt{60} = 6\sqrt{60}
\]
- Simplify \(\sqrt{60}\):
\[
\sqrt{60} = \sqrt{4 \cdot 15} = 2\sqrt{15}
\]
- Therefore:
\[
6\sqrt{60} = 6 \cdot 2\sqrt{15} = 12\sqrt{15}
\]
- Simplified Expression: \(12\sqrt{15}\)
- Color: Seafoam Green

---

Problem 3: \( 6\sqrt{126} + 3\sqrt{126} + 8\sqrt{126} \)

- Combine like terms:
\[
6\sqrt{126} + 3\sqrt{126} + 8\sqrt{126} = (6 + 3 + 8)\sqrt{126} = 17\sqrt{126}
\]
- Simplify \(\sqrt{126}\):
\[
\sqrt{126} = \sqrt{9 \cdot 14} = 3\sqrt{14}
\]
- Therefore:
\[
17\sqrt{126} = 17 \cdot 3\sqrt{14} = 51\sqrt{14}
\]
- Simplified Expression: \(51\sqrt{14}\)
- Color: Coral

---

Problem 4: \( 6\sqrt{54} - 5\sqrt{24} - 2\sqrt{10} \)

- Simplify each term:
\[
\sqrt{54} = \sqrt{9 \cdot 6} = 3\sqrt{6}, \quad \sqrt{24} = \sqrt{4 \cdot 6} = 2\sqrt{6}
\]
- Substitute back:
\[
6\sqrt{54} - 5\sqrt{24} - 2\sqrt{10} = 6(3\sqrt{6}) - 5(2\sqrt{6}) - 2\sqrt{10}
\]
\[
= 18\sqrt{6} - 10\sqrt{6} - 2\sqrt{10}
\]
- Combine like terms:
\[
18\sqrt{6} - 10\sqrt{6} = 8\sqrt{6}
\]
- Final expression:
\[
8\sqrt{6} - 2\sqrt{10}
\]
- Simplified Expression: \(8\sqrt{6} - 2\sqrt{10}\)
- Color: Sky Blue

---

Problem 5: \( -\sqrt[3]{320} - 4\sqrt[3]{5} + 2\sqrt[3]{135} + 2\sqrt[3]{16} \)

- Simplify each cube root:
\[
\sqrt[3]{320} = \sqrt[3]{64 \cdot 5} = 4\sqrt[3]{5}, \quad \sqrt[3]{135} = \sqrt[3]{27 \cdot 5} = 3\sqrt[3]{5}, \quad \sqrt[3]{16} = \sqrt[3]{8 \cdot 2} = 2\sqrt[3]{2}
\]
- Substitute back:
\[
-\sqrt[3]{320} - 4\sqrt[3]{5} + 2\sqrt[3]{135} + 2\sqrt[3]{16} = -4\sqrt[3]{5} - 4\sqrt[3]{5} + 2(3\sqrt[3]{5}) + 2(2\sqrt[3]{2})
\]
\[
= -4\sqrt[3]{5} - 4\sqrt[3]{5} + 6\sqrt[3]{5} + 4\sqrt[3]{2}
\]
- Combine like terms:
\[
-4\sqrt[3]{5} - 4\sqrt[3]{5} + 6\sqrt[3]{5} = -2\sqrt[3]{5}
\]
- Final expression:
\[
-2\sqrt[3]{5} + 4\sqrt[3]{2}
\]
- Simplified Expression: \(-2\sqrt[3]{5} + 4\sqrt[3]{2}\)
- Color: Gold

---

Problem 6: \( \sqrt{3} \cdot \sqrt{6} \)

- Use the property \(\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}\):
\[
\sqrt{3} \cdot \sqrt{6} = \sqrt{3 \cdot 6} = \sqrt{18}
\]
- Simplify \(\sqrt{18}\):
\[
\sqrt{18} = \sqrt{9 \cdot 2} = 3\sqrt{2}
\]
- Simplified Expression: \(3\sqrt{2}\)
- Color: Seafoam Green

---

Problem 7: \( \sqrt{6} \cdot \sqrt{2} \)

- Use the property \(\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}\):
\[
\sqrt{6} \cdot \sqrt{2} = \sqrt{6 \cdot 2} = \sqrt{12}
\]
- Simplify \(\sqrt{12}\):
\[
\sqrt{12} = \sqrt{4 \cdot 3} = 2\sqrt{3}
\]
- Simplified Expression: \(2\sqrt{3}\)
- Color: Lilac

---

Problem 8: \( 3\sqrt{3}(2 - \sqrt{5}) \)

- Distribute \(3\sqrt{3}\):
\[
3\sqrt{3}(2 - \sqrt{5}) = 3\sqrt{3} \cdot 2 - 3\sqrt{3} \cdot \sqrt{5}
\]
\[
= 6\sqrt{3} - 3\sqrt{15}
\]
- Simplified Expression: \(6\sqrt{3} - 3\sqrt{15}\)
- Color: Coral

---

Problem 9: \( 2\sqrt{15}(\sqrt{6} + 2\sqrt{5}) \)

- Distribute \(2\sqrt{15}\):
\[
2\sqrt{15}(\sqrt{6} + 2\sqrt{5}) = 2\sqrt{15} \cdot \sqrt{6} + 2\sqrt{15} \cdot 2\sqrt{5}
\]
\[
= 2\sqrt{90} + 4\sqrt{75}
\]
- Simplify each term:
\[
\sqrt{90} = \sqrt{9 \cdot 10} = 3\sqrt{10}, \quad \sqrt{75} = \sqrt{25 \cdot 3} = 5\sqrt{3}
\]
- Substitute back:
\[
2\sqrt{90} + 4\sqrt{75} = 2(3\sqrt{10}) + 4(5\sqrt{3})
\]
\[
= 6\sqrt{10} + 20\sqrt{3}
\]
- Simplified Expression: \(6\sqrt{10} + 20\sqrt{3}\)
- Color: Sky Blue

---

Problem 10: \( (\sqrt{2a} - 5)(5\sqrt{2a} - 3) \)

- Use the distributive property (FOIL):
\[
(\sqrt{2a} - 5)(5\sqrt{2a} - 3) = \sqrt{2a} \cdot 5\sqrt{2a} + \sqrt{2a} \cdot (-3) - 5 \cdot 5\sqrt{2a} - 5 \cdot (-3)
\]
\[
= 5(\sqrt{2a})^2 - 3\sqrt{2a} - 25\sqrt{2a} + 15
\]
- Simplify \((\sqrt{2a})^2\):
\[
(\sqrt{2a})^2 = 2a
\]
- Substitute back:
\[
5(2a) - 3\sqrt{2a} - 25\sqrt{2a} + 15 = 10a - 28\sqrt{2a} + 15
\]
- Simplified Expression: \(10a - 28\sqrt{2a} + 15\)
- Color: Gold

---

Problem 11: \( (7 + \sqrt{5})(1 + \sqrt{5}) \)

- Use the distributive property (FOIL):
\[
(7 + \sqrt{5})(1 + \sqrt{5}) = 7 \cdot 1 + 7 \cdot \sqrt{5} + \sqrt{5} \cdot 1 + \sqrt{5} \cdot \sqrt{5}
\]
\[
= 7 + 7\sqrt{5} + \sqrt{5} + (\sqrt{5})^2
\]
- Simplify \((\sqrt{5})^2\):
\[
(\sqrt{5})^2 = 5
\]
- Combine like terms:
\[
7 + 7\sqrt{5} + \sqrt{5} + 5 = 12 + 8\sqrt{5}
\]
- Simplified Expression: \(12 + 8\sqrt{5}\)
- Color: Sky Blue

---

Problem 12: \( \frac{\sqrt{16}}{\sqrt{25}} \)

- Simplify each square root:
\[
\sqrt{16} = 4, \quad \sqrt{25} = 5
\]
- Divide:
\[
\frac{\sqrt{16}}{\sqrt{25}} = \frac{4}{5}
\]
- Simplified Expression: \(\frac{4}{5}\)
- Color: Lilac

---

Problem 13: \( \frac{\sqrt{15}}{\sqrt{12}} \)

- Simplify the fraction:
\[
\frac{\sqrt{15}}{\sqrt{12}} = \sqrt{\frac{15}{12}} = \sqrt{\frac{5}{4}} = \frac{\sqrt{5}}{2}
\]
- Simplified Expression: \(\frac{\sqrt{5}}{2}\)
- Color: Coral

---

Problem 14: \( \frac{\sqrt{3}}{2\sqrt{5}} \)

- Rationalize the denominator:
\[
\frac{\sqrt{3}}{2\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}} = \frac{\sqrt{3} \cdot \sqrt{5}}{2 \cdot 5} = \frac{\sqrt{15}}{10}
\]
- Simplified Expression: \(\frac{\sqrt{15}}{10}\)
- Color: Seafoam Green

---

Problem 15: \( \frac{4\sqrt{2}}{3\sqrt{5}} \)

- Rationalize the denominator:
\[
\frac{4\sqrt{2}}{3\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}} = \frac{4\sqrt{2} \cdot \sqrt{5}}{3 \cdot 5} = \frac{4\sqrt{10}}{15}
\]
- Simplified Expression: \(\frac{4\sqrt{10}}{15}\)
- Color: Sky Blue

---

Problem 16: \( \frac{\sqrt{3}}{-1 - \sqrt{5}} \)

- Rationalize the denominator:
\[
\frac{\sqrt{3}}{-1 - \sqrt{5}} \cdot \frac{-1 + \sqrt{5}}{-1 + \sqrt{5}} = \frac{\sqrt{3}(-1 + \sqrt{5})}{(-1)^2 - (\sqrt{5})^2}
\]
\[
= \frac{\sqrt{3}(-1 + \sqrt{5})}{1 - 5} = \frac{\sqrt{3}(-1 + \sqrt{5})}{-4}
\]
- Simplify:
\[
= \frac{-\sqrt{3} + \sqrt{15}}{-4} = \frac{\sqrt{3} - \sqrt{15}}{4}
\]
- Simplified Expression: \(\frac{\sqrt{3} - \sqrt{15}}{4}\)
- Color: Gray

---

Final Answer:

\[
\boxed{
\begin{array}{ccc}
\text{Radical Expression} & \text{Simplified Expression} & \text{Color} \\
1. & -8\sqrt{3} & \text{lilac} \\
2. & 12\sqrt{15} & \text{seafoam green} \\
3. & 51\sqrt{14} & \text{coral} \\
4. & 8\sqrt{6} - 2\sqrt{10} & \text{sky blue} \\
5. & -2\sqrt[3]{5} + 4\sqrt[3]{2} & \text{gold} \\
6. & 3\sqrt{2} & \text{seafoam green} \\
7. & 2\sqrt{3} & \text{lilac} \\
8. & 6\sqrt{3} - 3\sqrt{15} & \text{coral} \\
9. & 6\sqrt{10} + 20\sqrt{3} & \text{sky blue} \\
10. & 10a - 28\sqrt{2a} + 15 & \text{gold} \\
11. & 12 + 8\sqrt{5} & \text{sky blue} \\
12. & \frac{4}{5} & \text{lilac} \\
13. & \frac{\sqrt{5}}{2} & \text{coral} \\
14. & \frac{\sqrt{15}}{10} & \text{seafoam green} \\
15. & \frac{4\sqrt{10}}{15} & \text{sky blue} \\
16. & \frac{\sqrt{3} - \sqrt{15}}{4} & \text{gray} \\
\end{array}
}
\]