This worksheet guides students through the fundamentals of simplifying radicals, from basic perfect squares to complex variable expressions.

Problem Overview:

The task involves simplifying radical expressions, specifically focusing on perfect squares and non-perfect squares. The goal is to simplify square roots by identifying perfect squares and reducing the expressions accordingly.

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Step-by-Step Solution:

#### Part 1: Perfect Squares
Perfect squares are numbers that can be expressed as the square of an integer. The sequence given is:
\[ 1, 4, 9, 16, 25, \_\_\_, \_\_\_, \_\_\_, \_\_\_, 144 \]

To find the missing terms:
- \( 1 = 1^2 \)
- \( 4 = 2^2 \)
- \( 9 = 3^2 \)
- \( 16 = 4^2 \)
- \( 25 = 5^2 \)
- Next: \( 6^2 = 36 \)
- Next: \( 7^2 = 49 \)
- Next: \( 8^2 = 64 \)
- Next: \( 9^2 = 81 \)
- Finally: \( 12^2 = 144 \)

So, the completed sequence is:
\[ 1, 4, 9, 16, 25, 36, 49, 64, 81, 144 \]

#### Exponents and Radicals
- \( x^2, x^4, x^6, \_\_\_ \): The pattern is increasing by 2. The next term is \( x^8 \).
- Exponents must be even for perfect squares.

#### Square Roots of Perfect Squares
- \( \sqrt{25} = 5 \) because \( 5^2 = 25 \)
- \( \sqrt{36} = 6 \) because \( 6^2 = 36 \)
- \( \sqrt{100} = 10 \) because \( 10^2 = 100 \)
- \( \sqrt{49} = 7 \) because \( 7^2 = 49 \)
- \( \sqrt{a^2} = a \) because \( (a)^2 = a^2 \)
- \( \sqrt{m^{16}} = m^8 \) because \( (m^8)^2 = m^{16} \)
- \( \sqrt{10^8} = 10^4 \) because \( (10^4)^2 = 10^8 \)
- \( \sqrt{8^2} = 8 \) because \( 8^2 = 64 \)

#### Simplifying Radicals
When simplifying radicals, we look for perfect square factors in the radicand and pull them out.

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Simplify (Simplifying Perfect Squares)

1. \( \sqrt{4} = 2 \) because \( 2^2 = 4 \)
2. \( \sqrt{16} = 4 \) because \( 4^2 = 16 \)
3. \( \sqrt{100} = 10 \) because \( 10^2 = 100 \)
4. \( \sqrt{a^8} = a^4 \) because \( (a^4)^2 = a^8 \)
5. \( \sqrt{w^{12}} = w^6 \) because \( (w^6)^2 = w^{12} \)
6. \( \sqrt{a^6b^{10}} = a^3b^5 \) because \( (a^3)^2 = a^6 \) and \( (b^5)^2 = b^{10} \)
7. \( \sqrt{9a^2} = 3a \) because \( \sqrt{9} = 3 \) and \( \sqrt{a^2} = a \)
8. \( -\sqrt{81m^{54}} = -9m^{27} \) because \( \sqrt{81} = 9 \) and \( \sqrt{m^{54}} = m^{27} \)
9. \( \sqrt{49a^8b^{12}} = 7a^4b^6 \) because \( \sqrt{49} = 7 \), \( \sqrt{a^8} = a^4 \), and \( \sqrt{b^{12}} = b^6 \)
10. \( \sqrt{121x^{14}y^6} = 11x^7y^3 \) because \( \sqrt{121} = 11 \), \( \sqrt{x^{14}} = x^7 \), and \( \sqrt{y^6} = y^3 \)

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Simplify (Simplifying Radicals that are not Perfect Squares)

1. \( \sqrt{20} = \sqrt{4 \cdot 5} = \sqrt{4} \cdot \sqrt{5} = 2\sqrt{5} \)
2. \( \sqrt{27} = \sqrt{9 \cdot 3} = \sqrt{9} \cdot \sqrt{3} = 3\sqrt{3} \)
3. \( \sqrt{48} = \sqrt{16 \cdot 3} = \sqrt{16} \cdot \sqrt{3} = 4\sqrt{3} \)
4. \( \sqrt{45} = \sqrt{9 \cdot 5} = \sqrt{9} \cdot \sqrt{5} = 3\sqrt{5} \)
5. \( \sqrt{12} = \sqrt{4 \cdot 3} = \sqrt{4} \cdot \sqrt{3} = 2\sqrt{3} \)
6. \( \sqrt{50} = \sqrt{25 \cdot 2} = \sqrt{25} \cdot \sqrt{2} = 5\sqrt{2} \)
7. \( \sqrt{a^{10}} = \sqrt{(a^5)^2} = a^5 \)
8. \( \sqrt{x^8} = \sqrt{(x^4)^2} = x^4 \)
9. \( \sqrt{x^3} = \sqrt{x^2 \cdot x} = \sqrt{x^2} \cdot \sqrt{x} = x\sqrt{x} \)

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Simplify

1. \( \sqrt{18} = \sqrt{9 \cdot 2} = \sqrt{9} \cdot \sqrt{2} = 3\sqrt{2} \)
2. \( \sqrt{125} = \sqrt{25 \cdot 5} = \sqrt{25} \cdot \sqrt{5} = 5\sqrt{5} \)
3. \( \sqrt{72} = \sqrt{36 \cdot 2} = \sqrt{36} \cdot \sqrt{2} = 6\sqrt{2} \)
4. \( \sqrt{180} = \sqrt{36 \cdot 5} = \sqrt{36} \cdot \sqrt{5} = 6\sqrt{5} \)
5. \( \sqrt{9^2} = 9 \)
6. \( \sqrt{b^7} = \sqrt{b^6 \cdot b} = \sqrt{b^6} \cdot \sqrt{b} = b^3\sqrt{b} \)
7. \( \sqrt{m^{11}} = \sqrt{m^{10} \cdot m} = \sqrt{m^{10}} \cdot \sqrt{m} = m^5\sqrt{m} \)
8. \( \sqrt{75x^3} = \sqrt{25 \cdot 3 \cdot x^2 \cdot x} = \sqrt{25} \cdot \sqrt{x^2} \cdot \sqrt{3x} = 5x\sqrt{3x} \)
9. \( \sqrt{27a^7b^4} = \sqrt{9 \cdot 3 \cdot a^6 \cdot a \cdot b^4} = \sqrt{9} \cdot \sqrt{a^6} \cdot \sqrt{b^4} \cdot \sqrt{3a} = 3a^3b^2\sqrt{3a} \)
10. \( \sqrt{32a^7b^4} = \sqrt{16 \cdot 2 \cdot a^6 \cdot a \cdot b^4} = \sqrt{16} \cdot \sqrt{a^6} \cdot \sqrt{b^4} \cdot \sqrt{2a} = 4a^3b^2\sqrt{2a} \)
11. \( \sqrt{9a^8} = \sqrt{9 \cdot a^8} = \sqrt{9} \cdot \sqrt{a^8} = 3a^4 \)
12. \( \sqrt{45a^7} = \sqrt{9 \cdot 5 \cdot a^6 \cdot a} = \sqrt{9} \cdot \sqrt{a^6} \cdot \sqrt{5a} = 3a^3\sqrt{5a} \)
13. \( \sqrt{36x^8y^6} = \sqrt{36 \cdot x^8 \cdot y^6} = \sqrt{36} \cdot \sqrt{x^8} \cdot \sqrt{y^6} = 6x^4y^3 \)
14. \( \sqrt{12x^{20}y^8} = \sqrt{4 \cdot 3 \cdot x^{20} \cdot y^8} = \sqrt{4} \cdot \sqrt{x^{20}} \cdot \sqrt{y^8} \cdot \sqrt{3} = 2x^{10}y^4\sqrt{3} \)
15. \( -\sqrt{200} = -\sqrt{100 \cdot 2} = -\sqrt{100} \cdot \sqrt{2} = -10\sqrt{2} \)
16. \( \sqrt{96} = \sqrt{16 \cdot 6} = \sqrt{16} \cdot \sqrt{6} = 4\sqrt{6} \)
17. \( \sqrt{63x^4} = \sqrt{9 \cdot 7 \cdot x^4} = \sqrt{9} \cdot \sqrt{x^4} \cdot \sqrt{7} = 3x^2\sqrt{7} \)
18. \( \sqrt{6x^2} = \sqrt{6 \cdot x^2} = \sqrt{x^2} \cdot \sqrt{6} = x\sqrt{6} \)
19. \( \sqrt{100x^2y^6} = \sqrt{100 \cdot x^2 \cdot y^6} = \sqrt{100} \cdot \sqrt{x^2} \cdot \sqrt{y^6} = 10xy^3 \)
20. \( \sqrt{80x^{20}y^9} = \sqrt{16 \cdot 5 \cdot x^{20} \cdot y^8 \cdot y} = \sqrt{16} \cdot \sqrt{x^{20}} \cdot \sqrt{y^8} \cdot \sqrt{5y} = 4x^{10}y^4\sqrt{5y} \)

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Final Answer

\[
\boxed{
\begin{aligned}
&\text{Perfect Squares: } 1, 4, 9, 16, 25, 36, 49, 64, 81, 144 \\
&\text{Exponents: } x^2, x^4, x^6, x^8 \\
&\text{Simplified Expressions: See detailed steps above.}
\end{aligned}
}
\]