Math worksheet for simplifying radicals, including problems on simplification, addition/subtraction, and multiplication of radical expressions.

Problem: Simplify Each Radical Expression to its Simplest Form

#### Simplify Each Radical Expression to its Simplest Form

1. $\sqrt{12x^{20}y^8}$
- Factor the radicand into perfect squares and other factors:
\[
\sqrt{12x^{20}y^8} = \sqrt{4 \cdot 3 \cdot x^{20} \cdot y^8}
\]
- Take the square root of the perfect squares:
\[
\sqrt{4} = 2, \quad \sqrt{x^{20}} = x^{10}, \quad \sqrt{y^8} = y^4
\]
- Combine the results:
\[
\sqrt{12x^{20}y^8} = 2x^{10}y^4\sqrt{3}
\]

2. $\sqrt{27x^{11}y^7}$
- Factor the radicand into perfect squares and other factors:
\[
\sqrt{27x^{11}y^7} = \sqrt{9 \cdot 3 \cdot x^{10} \cdot x \cdot y^6 \cdot y}
\]
- Take the square root of the perfect squares:
\[
\sqrt{9} = 3, \quad \sqrt{x^{10}} = x^5, \quad \sqrt{y^6} = y^3
\]
- Combine the results:
\[
\sqrt{27x^{11}y^7} = 3x^5y^3\sqrt{3xy}
\]

3. $\sqrt{80x^{100}y^{49}}$
- Factor the radicand into perfect squares and other factors:
\[
\sqrt{80x^{100}y^{49}} = \sqrt{16 \cdot 5 \cdot x^{100} \cdot y^{48} \cdot y}
\]
- Take the square root of the perfect squares:
\[
\sqrt{16} = 4, \quad \sqrt{x^{100}} = x^{50}, \quad \sqrt{y^{48}} = y^{24}
\]
- Combine the results:
\[
\sqrt{80x^{100}y^{49}} = 4x^{50}y^{24}\sqrt{5y}
\]

4. $\sqrt{20x^{16}y^5}$
- Factor the radicand into perfect squares and other factors:
\[
\sqrt{20x^{16}y^5} = \sqrt{4 \cdot 5 \cdot x^{16} \cdot y^4 \cdot y}
\]
- Take the square root of the perfect squares:
\[
\sqrt{4} = 2, \quad \sqrt{x^{16}} = x^8, \quad \sqrt{y^4} = y^2
\]
- Combine the results:
\[
\sqrt{20x^{16}y^5} = 2x^8y^2\sqrt{5y}
\]

5. $\sqrt{75x^8y^3}$
- Factor the radicand into perfect squares and other factors:
\[
\sqrt{75x^8y^3} = \sqrt{25 \cdot 3 \cdot x^8 \cdot y^2 \cdot y}
\]
- Take the square root of the perfect squares:
\[
\sqrt{25} = 5, \quad \sqrt{x^8} = x^4, \quad \sqrt{y^2} = y
\]
- Combine the results:
\[
\sqrt{75x^8y^3} = 5x^4y\sqrt{3y}
\]

6. $\sqrt{49a^8x^{12}}$
- Factor the radicand into perfect squares:
\[
\sqrt{49a^8x^{12}} = \sqrt{49 \cdot a^8 \cdot x^{12}}
\]
- Take the square root of the perfect squares:
\[
\sqrt{49} = 7, \quad \sqrt{a^8} = a^4, \quad \sqrt{x^{12}} = x^6
\]
- Combine the results:
\[
\sqrt{49a^8x^{12}} = 7a^4x^6
\]

Add or Subtract Radicals

7. $\sqrt{200} - \sqrt{72}$
- Simplify each radical:
\[
\sqrt{200} = \sqrt{100 \cdot 2} = 10\sqrt{2}
\]
\[
\sqrt{72} = \sqrt{36 \cdot 2} = 6\sqrt{2}
\]
- Combine like terms:
\[
\sqrt{200} - \sqrt{72} = 10\sqrt{2} - 6\sqrt{2} = 4\sqrt{2}
\]

8. $-6\sqrt{75} + 4\sqrt{125}$
- Simplify each radical:
\[
\sqrt{75} = \sqrt{25 \cdot 3} = 5\sqrt{3}
\]
\[
\sqrt{125} = \sqrt{25 \cdot 5} = 5\sqrt{5}
\]
- Substitute back:
\[
-6\sqrt{75} + 4\sqrt{125} = -6(5\sqrt{3}) + 4(5\sqrt{5}) = -30\sqrt{3} + 20\sqrt{5}
\]
- Since the radicals are not like terms, the expression is already simplified:
\[
-6\sqrt{75} + 4\sqrt{125} = -30\sqrt{3} + 20\sqrt{5}
\]

9. $3\sqrt{5} - \sqrt{x} + 4\sqrt{5} + 3\sqrt{x}$
- Combine like terms:
\[
3\sqrt{5} + 4\sqrt{5} = 7\sqrt{5}
\]
\[
-\sqrt{x} + 3\sqrt{x} = 2\sqrt{x}
\]
- Combine the results:
\[
3\sqrt{5} - \sqrt{x} + 4\sqrt{5} + 3\sqrt{x} = 7\sqrt{5} + 2\sqrt{x}
\]

10. $3 + 4\sqrt{x} - 6\sqrt{x}$
- Combine like terms:
\[
4\sqrt{x} - 6\sqrt{x} = -2\sqrt{x}
\]
- Combine the results:
\[
3 + 4\sqrt{x} - 6\sqrt{x} = 3 - 2\sqrt{x}
\]

Multiply and Simplify

11. $\sqrt[3]{(x+5)^2} \cdot \sqrt[3]{(x+5)^4}$
- Use the property of exponents for cube roots:
\[
\sqrt[3]{(x+5)^2} \cdot \sqrt[3]{(x+5)^4} = \sqrt[3]{(x+5)^{2+4}} = \sqrt[3]{(x+5)^6}
\]
- Simplify the cube root:
\[
\sqrt[3]{(x+5)^6} = (x+5)^2
\]

12. $\sqrt{15} \cdot \sqrt{15}$
- Use the property of square roots:
\[
\sqrt{15} \cdot \sqrt{15} = \sqrt{15 \cdot 15} = \sqrt{225}
\]
- Simplify the square root:
\[
\sqrt{225} = 15
\]

Final Answers

1. $\boxed{2x^{10}y^4\sqrt{3}}$
2. $\boxed{3x^5y^3\sqrt{3xy}}$
3. $\boxed{4x^{50}y^{24}\sqrt{5y}}$
4. $\boxed{2x^8y^2\sqrt{5y}}$
5. $\boxed{5x^4y\sqrt{3y}}$
6. $\boxed{7a^4x^6}$
7. $\boxed{4\sqrt{2}}$
8. $\boxed{-30\sqrt{3} + 20\sqrt{5}}$
9. $\boxed{7\sqrt{5} + 2\sqrt{x}}$
10. $\boxed{3 - 2\sqrt{x}}$
11. $\boxed{(x+5)^2}$
12. $\boxed{15}$