Simplifying Radical Expressions Coloring Worksheet Key

Problem Overview:

The task involves simplifying radical expressions and then using the simplified answers to color a picture according to a provided key. The worksheet lists 10 radical expressions, and each expression is associated with a specific color. The goal is to simplify each expression and match it with its corresponding color.

Step-by-Step Solution:

#### 1. Simplify Each Radical Expression:
We will simplify each radical expression step by step.

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Expression 1: \( \sqrt{36} \)
- \( \sqrt{36} = 6 \)
- Simplified Answer: \( 6 \)
- Color: LIGHT BLUE

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Expression 2: \( -\sqrt{81} \)
- \( \sqrt{81} = 9 \)
- \( -\sqrt{81} = -9 \)
- Simplified Answer: \( -9 \)
- Color: PINK

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Expression 3: \( \sqrt{50} \)
- Factorize 50 into prime factors: \( 50 = 2 \times 5^2 \)
- \( \sqrt{50} = \sqrt{2 \times 5^2} = 5\sqrt{2} \)
- Simplified Answer: \( 5\sqrt{2} \)
- Color: YELLOW

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Expression 4: \( \sqrt{200} \)
- Factorize 200 into prime factors: \( 200 = 2^3 \times 5^2 \)
- \( \sqrt{200} = \sqrt{2^3 \times 5^2} = \sqrt{2^2 \times 2 \times 5^2} = 2 \times 5 \sqrt{2} = 10\sqrt{2} \)
- Simplified Answer: \( 10\sqrt{2} \)
- Color: DARK BLUE

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Expression 5: \( 3\sqrt{20} \)
- Factorize 20 into prime factors: \( 20 = 2^2 \times 5 \)
- \( \sqrt{20} = \sqrt{2^2 \times 5} = 2\sqrt{5} \)
- \( 3\sqrt{20} = 3 \times 2\sqrt{5} = 6\sqrt{5} \)
- Simplified Answer: \( 6\sqrt{5} \)
- Color: RED

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Expression 6: \( \sqrt{75} \)
- Factorize 75 into prime factors: \( 75 = 3 \times 5^2 \)
- \( \sqrt{75} = \sqrt{3 \times 5^2} = 5\sqrt{3} \)
- Simplified Answer: \( 5\sqrt{3} \)
- Color: DARK BLUE

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Expression 7: \( -8\sqrt{24} \)
- Factorize 24 into prime factors: \( 24 = 2^3 \times 3 \)
- \( \sqrt{24} = \sqrt{2^3 \times 3} = \sqrt{2^2 \times 2 \times 3} = 2\sqrt{6} \)
- \( -8\sqrt{24} = -8 \times 2\sqrt{6} = -16\sqrt{6} \)
- Simplified Answer: \( -16\sqrt{6} \)
- Color: RED

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Expression 8: \( 5\sqrt{36} \)
- \( \sqrt{36} = 6 \)
- \( 5\sqrt{36} = 5 \times 6 = 30 \)
- Simplified Answer: \( 30 \)
- Color: LIGHT BLUE

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Expression 9: \( -6\sqrt{150} \)
- Factorize 150 into prime factors: \( 150 = 2 \times 3 \times 5^2 \)
- \( \sqrt{150} = \sqrt{2 \times 3 \times 5^2} = 5\sqrt{6} \)
- \( -6\sqrt{150} = -6 \times 5\sqrt{6} = -30\sqrt{6} \)
- Simplified Answer: \( -30\sqrt{6} \)
- Color: PINK

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Expression 10: \( -\sqrt{45} \)
- Factorize 45 into prime factors: \( 45 = 3^2 \times 5 \)
- \( \sqrt{45} = \sqrt{3^2 \times 5} = 3\sqrt{5} \)
- \( -\sqrt{45} = -3\sqrt{5} \)
- Simplified Answer: \( -3\sqrt{5} \)
- Color: YELLOW

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Final Answers and Colors:

1. \( \sqrt{36} = 6 \) → LIGHT BLUE
2. \( -\sqrt{81} = -9 \) → PINK
3. \( \sqrt{50} = 5\sqrt{2} \) → YELLOW
4. \( \sqrt{200} = 10\sqrt{2} \) → DARK BLUE
5. \( 3\sqrt{20} = 6\sqrt{5} \) → RED
6. \( \sqrt{75} = 5\sqrt{3} \) → DARK BLUE
7. \( -8\sqrt{24} = -16\sqrt{6} \) → RED
8. \( 5\sqrt{36} = 30 \) → LIGHT BLUE
9. \( -6\sqrt{150} = -30\sqrt{6} \) → PINK
10. \( -\sqrt{45} = -3\sqrt{5} \) → YELLOW

Final Answer:

\[
\boxed{
\begin{array}{ll}
1. & 6 \quad \text{(LIGHT BLUE)} \\
2. & -9 \quad \text{(PINK)} \\
3. & 5\sqrt{2} \quad \text{(YELLOW)} \\
4. & 10\sqrt{2} \quad \text{(DARK BLUE)} \\
5. & 6\sqrt{5} \quad \text{(RED)} \\
6. & 5\sqrt{3} \quad \text{(DARK BLUE)} \\
7. & -16\sqrt{6} \quad \text{(RED)} \\
8. & 30 \quad \text{(LIGHT BLUE)} \\
9. & -30\sqrt{6} \quad \text{(PINK)} \\
10. & -3\sqrt{5} \quad \text{(YELLOW)}
\end{array}
}
\]