Exponents and Radicals Worksheets | Exponents & Radicals ... - Free Printable

Problem: Simplify the Radical Expressions

We are tasked with simplifying each of the given radical expressions. Let's solve them step by step.

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#### 1) \( 5\sqrt{45b} \cdot 6\sqrt{48b} \)

- First, simplify the square roots:
\[
\sqrt{45b} = \sqrt{9 \cdot 5 \cdot b} = \sqrt{9} \cdot \sqrt{5b} = 3\sqrt{5b}
\]
\[
\sqrt{48b} = \sqrt{16 \cdot 3 \cdot b} = \sqrt{16} \cdot \sqrt{3b} = 4\sqrt{3b}
\]

- Now multiply the simplified terms:
\[
5\sqrt{45b} \cdot 6\sqrt{48b} = 5 \cdot 6 \cdot \sqrt{45b} \cdot \sqrt{48b}
\]
\[
= 30 \cdot \sqrt{45b} \cdot \sqrt{48b}
\]
\[
= 30 \cdot \sqrt{(45b)(48b)}
\]
\[
= 30 \cdot \sqrt{2160b^2}
\]

- Simplify \( \sqrt{2160b^2} \):
\[
\sqrt{2160b^2} = \sqrt{2160} \cdot \sqrt{b^2} = \sqrt{2160} \cdot b
\]
\[
\sqrt{2160} = \sqrt{36 \cdot 60} = \sqrt{36} \cdot \sqrt{60} = 6\sqrt{60}
\]
\[
\sqrt{60} = \sqrt{4 \cdot 15} = \sqrt{4} \cdot \sqrt{15} = 2\sqrt{15}
\]
\[
\sqrt{2160} = 6 \cdot 2\sqrt{15} = 12\sqrt{15}
\]

- Therefore:
\[
\sqrt{2160b^2} = 12\sqrt{15} \cdot b = 12b\sqrt{15}
\]

- Final expression:
\[
30 \cdot 12b\sqrt{15} = 360b\sqrt{15}
\]

Answer:
\[
\boxed{360b\sqrt{15}}
\]

---

#### 2) \( -\sqrt{45c} \left( -\sqrt{12c^2} + \sqrt{12c^3} \right) \)

- Simplify each term inside the parentheses:
\[
\sqrt{45c} = \sqrt{9 \cdot 5 \cdot c} = 3\sqrt{5c}
\]
\[
\sqrt{12c^2} = \sqrt{4 \cdot 3 \cdot c^2} = \sqrt{4} \cdot \sqrt{3} \cdot \sqrt{c^2} = 2c\sqrt{3}
\]
\[
\sqrt{12c^3} = \sqrt{4 \cdot 3 \cdot c^2 \cdot c} = \sqrt{4} \cdot \sqrt{3} \cdot \sqrt{c^2} \cdot \sqrt{c} = 2c\sqrt{3c}
\]

- Substitute back into the expression:
\[
-\sqrt{45c} \left( -\sqrt{12c^2} + \sqrt{12c^3} \right) = -3\sqrt{5c} \left( -2c\sqrt{3} + 2c\sqrt{3c} \right)
\]

- Distribute \( -3\sqrt{5c} \):
\[
= -3\sqrt{5c} \cdot (-2c\sqrt{3}) + (-3\sqrt{5c}) \cdot (2c\sqrt{3c})
\]
\[
= 6c\sqrt{5c} \cdot \sqrt{3} - 6c\sqrt{5c} \cdot \sqrt{3c}
\]
\[
= 6c\sqrt{15c} - 6c\sqrt{15c^2}
\]

- Simplify \( \sqrt{15c^2} \):
\[
\sqrt{15c^2} = \sqrt{15} \cdot \sqrt{c^2} = \sqrt{15} \cdot c = c\sqrt{15}
\]

- Substitute back:
\[
= 6c\sqrt{15c} - 6c \cdot c\sqrt{15}
\]
\[
= 6c\sqrt{15c} - 6c^2\sqrt{15}
\]

Answer:
\[
\boxed{6c\sqrt{15c} - 6c^2\sqrt{15}}
\]

---

#### 3) \( (7\sqrt{3d^2} + 2\sqrt{5}) \left( -4\sqrt{5d^2} - 5\sqrt{5} \right) \)

- Simplify each term:
\[
\sqrt{3d^2} = \sqrt{3} \cdot \sqrt{d^2} = d\sqrt{3}
\]
\[
\sqrt{5d^2} = \sqrt{5} \cdot \sqrt{d^2} = d\sqrt{5}
\]

- Substitute back into the expression:
\[
(7\sqrt{3d^2} + 2\sqrt{5}) \left( -4\sqrt{5d^2} - 5\sqrt{5} \right) = (7d\sqrt{3} + 2\sqrt{5}) \left( -4d\sqrt{5} - 5\sqrt{5} \right)
\]

- Use the distributive property (FOIL method):
\[
= (7d\sqrt{3})(-4d\sqrt{5}) + (7d\sqrt{3})(-5\sqrt{5}) + (2\sqrt{5})(-4d\sqrt{5}) + (2\sqrt{5})(-5\sqrt{5})
\]

- Simplify each term:
\[
(7d\sqrt{3})(-4d\sqrt{5}) = -28d^2\sqrt{15}
\]
\[
(7d\sqrt{3})(-5\sqrt{5}) = -35d\sqrt{15}
\]
\[
(2\sqrt{5})(-4d\sqrt{5}) = -8d(\sqrt{5})^2 = -8d \cdot 5 = -40d
\]
\[
(2\sqrt{5})(-5\sqrt{5}) = -10(\sqrt{5})^2 = -10 \cdot 5 = -50
\]

- Combine all terms:
\[
= -28d^2\sqrt{15} - 35d\sqrt{15} - 40d - 50
\]

Answer:
\[
\boxed{-28d^2\sqrt{15} - 35d\sqrt{15} - 40d - 50}
\]

---

#### 4) \( (-3\sqrt{7g^2} - 7\sqrt{2}) \left( 6\sqrt{7g^2} + \sqrt{2} \right) \)

- Simplify each term:
\[
\sqrt{7g^2} = \sqrt{7} \cdot \sqrt{g^2} = g\sqrt{7}
\]

- Substitute back into the expression:
\[
(-3\sqrt{7g^2} - 7\sqrt{2}) \left( 6\sqrt{7g^2} + \sqrt{2} \right) = (-3g\sqrt{7} - 7\sqrt{2}) \left( 6g\sqrt{7} + \sqrt{2} \right)
\]

- Use the distributive property (FOIL method):
\[
= (-3g\sqrt{7})(6g\sqrt{7}) + (-3g\sqrt{7})(\sqrt{2}) + (-7\sqrt{2})(6g\sqrt{7}) + (-7\sqrt{2})(\sqrt{2})
\]

- Simplify each term:
\[
(-3g\sqrt{7})(6g\sqrt{7}) = -18g^2(\sqrt{7})^2 = -18g^2 \cdot 7 = -126g^2
\]
\[
(-3g\sqrt{7})(\sqrt{2}) = -3g\sqrt{14}
\]
\[
(-7\sqrt{2})(6g\sqrt{7}) = -42g\sqrt{14}
\]
\[
(-7\sqrt{2})(\sqrt{2}) = -7(\sqrt{2})^2 = -7 \cdot 2 = -14
\]

- Combine all terms:
\[
= -126g^2 - 3g\sqrt{14} - 42g\sqrt{14} - 14
\]
\[
= -126g^2 - 45g\sqrt{14} - 14
\]

Answer:
\[
\boxed{-126g^2 - 45g\sqrt{14} - 14}
\]

---

#### 5) \( -4\sqrt{63} \left( 5\sqrt{48} + \sqrt{6} \right) \)

- Simplify each term:
\[
\sqrt{63} = \sqrt{9 \cdot 7} = \sqrt{9} \cdot \sqrt{7} = 3\sqrt{7}
\]
\[
\sqrt{48} = \sqrt{16 \cdot 3} = \sqrt{16} \cdot \sqrt{3} = 4\sqrt{3}
\]

- Substitute back into the expression:
\[
-4\sqrt{63} \left( 5\sqrt{48} + \sqrt{6} \right) = -4(3\sqrt{7}) \left( 5(4\sqrt{3}) + \sqrt{6} \right)
\]
\[
= -12\sqrt{7} \left( 20\sqrt{3} + \sqrt{6} \right)
\]

- Distribute \( -12\sqrt{7} \):
\[
= -12\sqrt{7} \cdot 20\sqrt{3} + (-12\sqrt{7}) \cdot \sqrt{6}
\]
\[
= -240\sqrt{21} - 12\sqrt{42}
\]

Answer:
\[
\boxed{-240\sqrt{21} - 12\sqrt{42}}
\]

---

#### 6) \( -6\sqrt{8} \cdot 7\sqrt{63} \)

- Simplify each term:
\[
\sqrt{8} = \sqrt{4 \cdot 2} = \sqrt{4} \cdot \sqrt{2} = 2\sqrt{2}
\]
\[
\sqrt{63} = \sqrt{9 \cdot 7} = \sqrt{9} \cdot \sqrt{7} = 3\sqrt{7}
\]

- Multiply the simplified terms:
\[
-6\sqrt{8} \cdot 7\sqrt{63} = -6 \cdot 7 \cdot \sqrt{8} \cdot \sqrt{63}
\]
\[
= -42 \cdot \sqrt{8} \cdot \sqrt{63}
\]
\[
= -42 \cdot \sqrt{(8)(63)}
\]
\[
= -42 \cdot \sqrt{504}
\]

- Simplify \( \sqrt{504} \):
\[
\sqrt{504} = \sqrt{36 \cdot 14} = \sqrt{36} \cdot \sqrt{14} = 6\sqrt{14}
\]

- Therefore:
\[
-42 \cdot \sqrt{504} = -42 \cdot 6\sqrt{14} = -252\sqrt{14}
\]

Answer:
\[
\boxed{-252\sqrt{14}}
\]

---

#### 7) \( \sqrt{32n} \left( \sqrt{20n^2} - \sqrt{99n^3} \right) \)

- Simplify each term:
\[
\sqrt{32n} = \sqrt{16 \cdot 2 \cdot n} = \sqrt{16} \cdot \sqrt{2n} = 4\sqrt{2n}
\]
\[
\sqrt{20n^2} = \sqrt{4 \cdot 5 \cdot n^2} = \sqrt{4} \cdot \sqrt{5} \cdot \sqrt{n^2} = 2n\sqrt{5}
\]
\[
\sqrt{99n^3} = \sqrt{9 \cdot 11 \cdot n^2 \cdot n} = \sqrt{9} \cdot \sqrt{11} \cdot \sqrt{n^2} \cdot \sqrt{n} = 3n\sqrt{11n}
\]

- Substitute back into the expression:
\[
\sqrt{32n} \left( \sqrt{20n^2} - \sqrt{99n^3} \right) = 4\sqrt{2n} \left( 2n\sqrt{5} - 3n\sqrt{11n} \right)
\]

- Distribute \( 4\sqrt{2n} \):
\[
= 4\sqrt{2n} \cdot 2n\sqrt{5} - 4\sqrt{2n} \cdot 3n\sqrt{11n}
\]
\[
= 8n\sqrt{10n} - 12n\sqrt{22n^2}
\]

- Simplify \( \sqrt{22n^2} \):
\[
\sqrt{22n^2} = \sqrt{22} \cdot \sqrt{n^2} = \sqrt{22} \cdot n = n\sqrt{22}
\]

- Substitute back:
\[
= 8n\sqrt{10n} - 12n \cdot n\sqrt{22}
\]
\[
= 8n\sqrt{10n} - 12n^2\sqrt{22}
\]

Answer:
\[
\boxed{8n\sqrt{10n} - 12n^2\sqrt{22}}
\]

---

#### 8) \( -\sqrt{112p} \cdot \sqrt{44p} \)

- Simplify each term:
\[
\sqrt{112p} = \sqrt{16 \cdot 7 \cdot p} = \sqrt{16} \cdot \sqrt{7p} = 4\sqrt{7p}
\]
\[
\sqrt{44p} = \sqrt{4 \cdot 11 \cdot p} = \sqrt{4} \cdot \sqrt{11p} = 2\sqrt{11p}
\]

- Multiply the simplified terms:
\[
-\sqrt{112p} \cdot \sqrt{44p} = -(4\sqrt{7p}) \cdot (2\sqrt{11p})
\]
\[
= -4 \cdot 2 \cdot \sqrt{7p} \cdot \sqrt{11p}
\]
\[
= -8 \cdot \sqrt{(7p)(11p)}
\]
\[
= -8 \cdot \sqrt{77p^2}
\]

- Simplify \( \sqrt{77p^2} \):
\[
\sqrt{77p^2} = \sqrt{77} \cdot \sqrt{p^2} = \sqrt{77} \cdot p = p\sqrt{77}
\]

- Therefore:
\[
-8 \cdot \sqrt{77p^2} = -8 \cdot p\sqrt{77} = -8p\sqrt{77}
\]

Answer:
\[
\boxed{-8p\sqrt{77}}
\]

---

#### 9) \( (-2\sqrt{3} + 6\sqrt{11}) \left( 3\sqrt{3} - 6\sqrt{11} \right) \)

- Use the distributive property (FOIL method):
\[
(-2\sqrt{3} + 6\sqrt{11}) \left( 3\sqrt{3} - 6\sqrt{11} \right)
\]
\[
= (-2\sqrt{3})(3\sqrt{3}) + (-2\sqrt{3})(-6\sqrt{11}) + (6\sqrt{11})(3\sqrt{3}) + (6\sqrt{11})(-6\sqrt{11})
\]

- Simplify each term:
\[
(-2\sqrt{3})(3\sqrt{3}) = -6(\sqrt{3})^2 = -6 \cdot 3 = -18
\]
\[
(-2\sqrt{3})(-6\sqrt{11}) = 12\sqrt{33}
\]
\[
(6\sqrt{11})(3\sqrt{3}) = 18\sqrt{33}
\]
\[
(6\sqrt{11})(-6\sqrt{11}) = -36(\sqrt{11})^2 = -36 \cdot 11 = -396
\]

- Combine all terms:
\[
= -18 + 12\sqrt{33} + 18\sqrt{33} - 396
\]
\[
= -18 - 396 + (12\sqrt{33} + 18\sqrt{33})
\]
\[
= -414 + 30\sqrt{33}
\]

Answer:
\[
\boxed{-414 + 30\sqrt{33}}
\]

---

#### 10) \( (-\sqrt{7} + \sqrt{11}) \left( -\sqrt{7} + \sqrt{11} \right) \)

- This is a perfect square trinomial:
\[
(-\sqrt{7} + \sqrt{11}) \left( -\sqrt{7} + \sqrt{11} \right) = \left( -\sqrt{7} + \sqrt{11} \right)^2
\]

- Expand using the formula \( (a - b)^2 = a^2 - 2ab + b^2 \):
\[
\left( -\sqrt{7} + \sqrt{11} \right)^2 = (-\sqrt{7})^2 - 2(-\sqrt{7})(\sqrt{11}) + (\sqrt{11})^2
\]

- Simplify each term:
\[
(-\sqrt{7})^2 = (\sqrt{7})^2 = 7
\]
\[
-2(-\sqrt{7})(\sqrt{11}) = 2\sqrt{77}
\]
\[
(\sqrt{11})^2 = 11
\]

- Combine all terms:
\[
= 7 + 2\sqrt{77} + 11
\]
\[
= 18 + 2\sqrt{77}
\]

Answer:
\[
\boxed{18 + 2\sqrt{77}}
\]

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Final Answers:

1. \(\boxed{360b\sqrt{15}}\)
2. \(\boxed{6c\sqrt{15c} - 6c^2\sqrt{15}}\)
3. \(\boxed{-28d^2\sqrt{15} - 35d\sqrt{15} - 40d - 50}\)
4. \(\boxed{-126g^2 - 45g\sqrt{14} - 14}\)
5. \(\boxed{-240\sqrt{21} - 12\sqrt{42}}\)
6. \(\boxed{-252\sqrt{14}}\)
7. \(\boxed{8n\sqrt{10n} - 12n^2\sqrt{22}}\)
8. \(\boxed{-8p\sqrt{77}}\)
9. \(\boxed{-414 + 30\sqrt{33}}\)
10. \(\boxed{18 + 2\sqrt{77}}\)