Question

simplify to simplest radical form. show all valid and appropriate work on your paper.

1. \\(\sqrt{36}\\)
2. \\(\sqrt{24}\\)
3. \\(\sqrt{60}\\)
4. \\(-\sqrt{126}\\)
5. \\(\sqrt{147}\\)
6. \\(\sqrt{216}\\)
7. \\(\sqrt{324}\\)
8. \\(-\sqrt{600}\\)
9. \\(4\sqrt{20}\\)
10. \\(-6\sqrt{54}\\)
11. \\(7\sqrt{64}\\)
12. \\(9\sqrt{120}\\)
13. \\(\frac{3}{4}\sqrt{32}\\)
14. \\(-\frac{5}{3}\sqrt{27}\\)
15. \\(\frac{3}{16}\sqrt{180}\\)
16. \\(5\sqrt{3} + 3\sqrt{3}\\)
17. \\(7\sqrt{6} - \sqrt{6}\\)
18. \\(2\sqrt{5} - 9\sqrt{5}\\)
19. \\(4\sqrt{2} - \sqrt{32}\\)
20. \\(\sqrt{27} - \sqrt{3}\\)
21. \\(5\sqrt{18} - 2\sqrt{50}\\)

Explanation:

Step1: Factor into perfect square

$\sqrt{36} = \sqrt{6^2}$

Step2: Simplify the radical

$\sqrt{6^2} = 6$

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Step1: Factor into perfect square + remainder

$\sqrt{24} = \sqrt{4 \times 6}$

Step2: Split the radical

$\sqrt{4 \times 6} = \sqrt{4} \times \sqrt{6}$

Step3: Simplify perfect square radical

$\sqrt{4} \times \sqrt{6} = 2\sqrt{6}$

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Step1: Factor into perfect square + remainder

$\sqrt{60} = \sqrt{4 \times 15}$

Step2: Split the radical

$\sqrt{4 \times 15} = \sqrt{4} \times \sqrt{15}$

Step3: Simplify perfect square radical

$\sqrt{4} \times \sqrt{15} = 2\sqrt{15}$

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Step1: Factor into perfect square + remainder

$-\sqrt{126} = -\sqrt{9 \times 14}$

Step2: Split the radical

$-\sqrt{9 \times 14} = -\sqrt{9} \times \sqrt{14}$

Step3: Simplify perfect square radical

$-\sqrt{9} \times \sqrt{14} = -3\sqrt{14}$

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Step1: Factor into perfect square + remainder

$\sqrt{147} = \sqrt{49 \times 3}$

Step2: Split the radical

$\sqrt{49 \times 3} = \sqrt{49} \times \sqrt{3}$

Step3: Simplify perfect square radical

$\sqrt{49} \times \sqrt{3} = 7\sqrt{3}$

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Step1: Factor into perfect square + remainder

$\sqrt{216} = \sqrt{36 \times 6}$

Step2: Split the radical

$\sqrt{36 \times 6} = \sqrt{36} \times \sqrt{6}$

Step3: Simplify perfect square radical

$\sqrt{36} \times \sqrt{6} = 6\sqrt{6}$

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Step1: Factor into perfect square

$\sqrt{324} = \sqrt{18^2}$

Step2: Simplify the radical

$\sqrt{18^2} = 18$

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Step1: Factor into perfect square + remainder

$-\sqrt{600} = -\sqrt{100 \times 6}$

Step2: Split the radical

$-\sqrt{100 \times 6} = -\sqrt{100} \times \sqrt{6}$

Step3: Simplify perfect square radical

$-\sqrt{100} \times \sqrt{6} = -10\sqrt{6}$

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Step1: Factor into perfect square + remainder

$4\sqrt{20} = 4\sqrt{4 \times 5}$

Step2: Split the radical

$4\sqrt{4 \times 5} = 4 \times \sqrt{4} \times \sqrt{5}$

Step3: Simplify and multiply

$4 \times 2 \times \sqrt{5} = 8\sqrt{5}$

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Step1: Factor into perfect square + remainder

$-6\sqrt{54} = -6\sqrt{9 \times 6}$

Step2: Split the radical

$-6\sqrt{9 \times 6} = -6 \times \sqrt{9} \times \sqrt{6}$

Step3: Simplify and multiply

$-6 \times 3 \times \sqrt{6} = -18\sqrt{6}$

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Step1: Factor into perfect square

$7\sqrt{64} = 7\sqrt{8^2}$

Step2: Simplify and multiply

$7 \times 8 = 56$

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Step1: Factor into perfect square + remainder

$9\sqrt{120} = 9\sqrt{4 \times 30}$

Step2: Split the radical

$9\sqrt{4 \times 30} = 9 \times \sqrt{4} \times \sqrt{30}$

Step3: Simplify and multiply

$9 \times 2 \times \sqrt{30} = 18\sqrt{30}$

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Step1: Factor into perfect square + remainder

$\frac{3}{4}\sqrt{32} = \frac{3}{4}\sqrt{16 \times 2}$

Step2: Split the radical

$\frac{3}{4}\sqrt{16 \times 2} = \frac{3}{4} \times \sqrt{16} \times \sqrt{2}$

Step3: Simplify and multiply

$\frac{3}{4} \times 4 \times \sqrt{2} = 3\sqrt{2}$

---

Step1: Factor into perfect square + remainder

$-\frac{5}{3}\sqrt{27} = -\frac{5}{3}\sqrt{9 \times 3}$

Step2: Split the radical

$-\frac{5}{3}\sqrt{9 \times 3} = -\frac{5}{3} \times \sqrt{9} \times \sqrt{3}$

Step3: Simplify and multiply

$-\frac{5}{3} \times 3 \times \sqrt{3} = -5\sqrt{3}$

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Step1: Factor into perfect square + remainder

$\frac{3}{16}\sqrt{180} = \frac{3}{16}\sqrt{36 \times 5}$

Step2: Split the radical

$\frac{3}{16}\sqrt{36 \times 5} = \frac{3}{16} \times \sqrt{36} \times \sqrt{5}$

Step3: Simplify and multiply

$\frac{3}{16} \times 6 \times \sqrt{5} = \frac{18}{16}\sqrt{5} = \frac{9}{8}\sqrt{5}$

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Step1: Combine like radicals

$5\sqrt{3} + 3\sqrt{3} = (5+3)\sqrt{3}$

Step2: Simplify the coefficient

$(5+3)\sqrt{3} = 8\sqrt{3}$

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Step1: Combine like radicals

$7\sqrt{6} - \sqrt{6} = (7-1)\sqrt{6}$

Step2: Simplify the coefficient

$(7-1)\sqrt{6} = 6\sqrt{6}$

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Step1: Combine like radicals

$2\sqrt{5} - 9\sqrt{5} = (2-9)\sqrt{5}$

Step2: Simplify the coefficient

$(2-9)\sqrt{5} = -7\sqrt{5}$

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Step1: Simplify $\sqrt{32}$ to like radical

$\sqrt{32} = \sqrt{16 \times 2} = 4\sqrt{2}$

Step2: Subtract like radicals

$4\sqrt{2} - 4\sqrt{2} = (4-4)\sqrt{2}$

Step3: Simplify the coefficient

$(4-4)\sqrt{2} = 0$

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Step1: Simplify $\sqrt{27}$ to like radical

$\sqrt{27} = \sqrt{9 \times 3} = 3\sqrt{3}$

Step2: Subtract like radicals

$3\sqrt{3} - \sqrt{3} = (3-1)\sqrt{3}$

Step3: Simplify the coefficient

$(3-1)\sqrt{3} = 2\sqrt{3}$

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Step1: Simplify both radicals to like terms

$5\sqrt{18} = 5\sqrt{9 \times 2} = 15\sqrt{2}$; $2\sqrt{50} = 2\sqrt{25 \times 2} = 10\sqrt{2}$

Step2: Subtract like radicals

$15\sqrt{2} - 10\sqrt{2} = (15-10)\sqrt{2}$

Step3: Simplify the coefficient

$(15-10)\sqrt{2} = 5\sqrt{2}$

Answer:

1. $6$
2. $2\sqrt{6}$
3. $2\sqrt{15}$
4. $-3\sqrt{14}$
5. $7\sqrt{3}$
6. $6\sqrt{6}$
7. $18$
8. $-10\sqrt{6}$
9. $8\sqrt{5}$
10. $-18\sqrt{6}$
11. $56$
12. $18\sqrt{30}$
13. $3\sqrt{2}$
14. $-5\sqrt{3}$
15. $\frac{9}{8}\sqrt{5}$
16. $8\sqrt{3}$
17. $6\sqrt{6}$
18. $-7\sqrt{5}$
19. $0$
20. $2\sqrt{3}$
21. $5\sqrt{2}$