4. Simplify. $$ \begin{array}{llll} ext { a) } 2 \sqrt{6} imes 5 \sqrt{3} & ext { b) }(-6 \sqrt{6})(5 \sqrt{12}) & ext { c) } 3 \sqrt{6} imes 2 \sqrt{18} imes \sqrt{15} & ext { d) }(5 \sqrt{7}-2 \sqrt{3})(\sqrt{7}+4 \sqrt{3})\end{array} $$

Question

4. Simplify. $$ \begin{array}{llll}	ext { a) } 2 \sqrt{6} 	imes 5 \sqrt{3} & 	ext { b) }(-6 \sqrt{6})(5 \sqrt{12}) & 	ext { c) } 3 \sqrt{6} 	imes 2 \sqrt{18} 	imes \sqrt{15} & 	ext { d) }(5 \sqrt{7}-2 \sqrt{3})(\sqrt{7}+4 \sqrt{3})\end{array} $$

UpStudy AI · Accepted Answer

**a) Simplify $$2 \sqrt{6} \times 5 \sqrt{3}$$**

1. Multiply the coefficients:
   $$
   2 \times 5 = 10.
   $$
2. Multiply the radicals using the property $$\sqrt{a}\cdot\sqrt{b}=\sqrt{ab}$$:
   $$
   \sqrt{6} \times \sqrt{3} = \sqrt{18}.
   $$
3. Simplify the radical:
   $$
   \sqrt{18} = \sqrt{9 \cdot 2} = 3 \sqrt{2}.
   $$
4. Multiply the coefficient with the simplified radical:
   $$
   10 \times 3 \sqrt{2} = 30 \sqrt{2}.
   $$

Thus, the answer is $$30 \sqrt{2}$$.

---

**b) Simplify $$(-6 \sqrt{6})(5 \sqrt{12})$$**

1. Multiply the coefficients:
   $$
   -6 \times 5 = -30.
   $$
2. Multiply the radicals:
   $$
   \sqrt{6} \times \sqrt{12} = \sqrt{72}.
   $$
3. Simplify the radical:
   $$
   \sqrt{72} = \sqrt{36 \cdot 2} = 6 \sqrt{2}.
   $$
4. Multiply the coefficient with the simplified radical:
   $$
   -30 \times 6 \sqrt{2} = -180 \sqrt{2}.
   $$

Thus, the answer is $$-180 \sqrt{2}$$.

---

**c) Simplify $$3 \sqrt{6} \times 2 \sqrt{18} \times \sqrt{15}$$**

1. Multiply the coefficients:
   $$
   3 \times 2 = 6.
   $$
2. Multiply the radicals:
   $$
   \sqrt{6} \times \sqrt{18} \times \sqrt{15} = \sqrt{6 \times 18 \times 15}.
   $$
3. Compute the product under the radical:
   $$
   6 \times 18 \times 15 = 1620.
   $$
4. Factor $$1620$$ to simplify:
   $$
   1620 = 81 \times 20,\quad \sqrt{1620} = \sqrt{81 \times 20} = 9 \sqrt{20}.
   $$
5. Notice that $$\sqrt{20}$$ can be simplified:
   $$
   \sqrt{20} = \sqrt{4 \times 5} = 2 \sqrt{5}.
   $$
6. Combining the above:
   $$
   9 \sqrt{20} = 9 \times 2 \sqrt{5} = 18 \sqrt{5}.
   $$
7. Multiply with the previously obtained coefficient:
   $$
   6 \times 18 \sqrt{5} = 108 \sqrt{5}.
   $$

Thus, the answer is $$108 \sqrt{5}$$.

---

**d) Simplify $$(5 \sqrt{7}-2 \sqrt{3})(\sqrt{7}+4 \sqrt{3})$$**

1. Distribute using the FOIL method:

- First terms: 
     $$
     5 \sqrt{7} \times \sqrt{7} = 5 \times 7 = 35.
     $$
   - Outer terms:
     $$
     5 \sqrt{7} \times 4 \sqrt{3} = 20 \sqrt{21}.
     $$
   - Inner terms:
     $$
     -2 \sqrt{3} \times \sqrt{7} = -2 \sqrt{21}.
     $$
   - Last terms:
     $$
     -2 \sqrt{3} \times 4 \sqrt{3} = -8 \times 3 = -24.
     $$

2. Combine like terms:
   - Combine the constant terms:
     $$
     35 - 24 = 11.
     $$
   - Combine the radical terms:
     $$
     20 \sqrt{21} - 2 \sqrt{21} = 18 \sqrt{21}.
     $$

Thus, the answer is $$11 + 18 \sqrt{21}$$.
a) $$30 \sqrt{2}$$ b) $$-180 \sqrt{2}$$ c) $$108 \sqrt{5}$$ d) $$11 + 18 \sqrt{21}$$

Simplify.

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