Grade 9 math worksheet focused on simplifying radicals, including problems for simplifying square roots and combining like radical terms.

Here are the step-by-step solutions for the problems on the worksheet.

A. Simplify

To simplify a square root, look for the largest perfect square (like 4, 9, 16, 25, 36...) that divides evenly into the number. Split the number into that perfect square and the remaining factor, then take the square root of the perfect square.

1) $\sqrt{28}$
* Factors: $4 \times 7$
* $\sqrt{4} \times \sqrt{7} = 2\sqrt{7}$

2) $\sqrt{72}$
* Factors: $36 \times 2$
* $\sqrt{36} \times \sqrt{2} = 6\sqrt{2}$

3) $\sqrt{32}$
* Factors: $16 \times 2$
* $\sqrt{16} \times \sqrt{2} = 4\sqrt{2}$

4) $\sqrt{288}$
* Factors: $144 \times 2$
* $\sqrt{144} \times \sqrt{2} = 12\sqrt{2}$

5) $\sqrt{252}$
* Factors: $36 \times 7$
* $\sqrt{36} \times \sqrt{7} = 6\sqrt{7}$

6) $\sqrt{98}$
* Factors: $49 \times 2$
* $\sqrt{49} \times \sqrt{2} = 7\sqrt{2}$

7) $\sqrt{45}$
* Factors: $9 \times 5$
* $\sqrt{9} \times \sqrt{5} = 3\sqrt{5}$

8) $\sqrt{448}$
* Factors: $64 \times 7$
* $\sqrt{64} \times \sqrt{7} = 8\sqrt{7}$

9) $\sqrt{150}$
* Factors: $25 \times 6$
* $\sqrt{25} \times \sqrt{6} = 5\sqrt{6}$

10) $\sqrt{384}$
* Factors: $64 \times 6$
* $\sqrt{64} \times \sqrt{6} = 8\sqrt{6}$

11) $-\sqrt{50}$
* Factors inside: $25 \times 2$
* $-(\sqrt{25} \times \sqrt{2}) = -5\sqrt{2}$

12) $2\sqrt{144}$
* $\sqrt{144} = 12$
* $2 \times 12 = 24$

13) $-3\sqrt{80}$
* Factors inside: $16 \times 5$
* $-3(\sqrt{16} \times \sqrt{5}) = -3(4\sqrt{5}) = -12\sqrt{5}$

14) $3\sqrt{36}$
* $\sqrt{36} = 6$
* $3 \times 6 = 18$

15) $5\sqrt{75}$
* Factors inside: $25 \times 3$
* $5(\sqrt{25} \times \sqrt{3}) = 5(5\sqrt{3}) = 25\sqrt{3}$

16) $-4\sqrt{384}$
* From #10, we know $\sqrt{384} = 8\sqrt{6}$
* $-4(8\sqrt{6}) = -32\sqrt{6}$

17) $-\sqrt{245}$
* Factors inside: $49 \times 5$
* $-(\sqrt{49} \times \sqrt{5}) = -7\sqrt{5}$

18) $6\sqrt{36}$
* $\sqrt{36} = 6$
* $6 \times 6 = 36$

19) $3\sqrt{98}$
* From #6, we know $\sqrt{98} = 7\sqrt{2}$
* $3(7\sqrt{2}) = 21\sqrt{2}$

20) $6\sqrt{12}$
* Factors inside: $4 \times 3$
* $6(\sqrt{4} \times \sqrt{3}) = 6(2\sqrt{3}) = 12\sqrt{3}$

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B. Simplify (Add/Subt)

First, simplify each radical individually. Then, combine terms that have the same number under the square root symbol (like combining $x$'s in algebra).

21) $\sqrt{8} + \sqrt{8}$
* $\sqrt{8} = 2\sqrt{2}$
* $2\sqrt{2} + 2\sqrt{2} = 4\sqrt{2}$

22) $\sqrt{45} + \sqrt{20}$
* $\sqrt{45} = 3\sqrt{5}$
* $\sqrt{20} = 2\sqrt{5}$
* $3\sqrt{5} + 2\sqrt{5} = 5\sqrt{5}$

23) $\sqrt{5} + \sqrt{5}$
* $1\sqrt{5} + 1\sqrt{5} = 2\sqrt{5}$

24) $\sqrt{45} + \sqrt{5}$
* $3\sqrt{5} + 1\sqrt{5} = 4\sqrt{5}$

25) $\sqrt{2} + \sqrt{2}$
* $1\sqrt{2} + 1\sqrt{2} = 2\sqrt{2}$

26) $\sqrt{6} + \sqrt{2} + \sqrt{2}$
* Combine the $\sqrt{2}$'s: $\sqrt{6} + 2\sqrt{2}$ (Cannot combine further)

27) $\sqrt{6} + \sqrt{54} + \sqrt{6}$
* $\sqrt{54} = \sqrt{9 \times 6} = 3\sqrt{6}$
* $\sqrt{6} + 3\sqrt{6} + \sqrt{6} = 5\sqrt{6}$

28) $\sqrt{45} + \sqrt{24} + \sqrt{45}$
* $\sqrt{45} = 3\sqrt{5}$
* $\sqrt{24} = \sqrt{4 \times 6} = 2\sqrt{6}$
* $3\sqrt{5} + 2\sqrt{6} + 3\sqrt{5} = 6\sqrt{5} + 2\sqrt{6}$

29) $\sqrt{54} + \sqrt{5} + \sqrt{6}$
* $\sqrt{54} = 3\sqrt{6}$
* $3\sqrt{6} + \sqrt{5} + \sqrt{6} = 4\sqrt{6} + \sqrt{5}$

30) $\sqrt{27} + \sqrt{18} + \sqrt{8}$
* $\sqrt{27} = 3\sqrt{3}$
* $\sqrt{18} = 3\sqrt{2}$
* $\sqrt{8} = 2\sqrt{2}$
* $3\sqrt{3} + 3\sqrt{2} + 2\sqrt{2} = 3\sqrt{3} + 5\sqrt{2}$

31) $\sqrt{18} + \sqrt{6} + \sqrt{8} + \sqrt{27}$
* $\sqrt{18} = 3\sqrt{2}$
* $\sqrt{8} = 2\sqrt{2}$
* $\sqrt{27} = 3\sqrt{3}$
* Combine $\sqrt{2}$'s: $3\sqrt{2} + 2\sqrt{2} = 5\sqrt{2}$
* Result: $5\sqrt{2} + \sqrt{6} + 3\sqrt{3}$

32) $\sqrt{18} + \sqrt{2} + \sqrt{20} + \sqrt{2}$
* $\sqrt{18} = 3\sqrt{2}$
* $\sqrt{20} = 2\sqrt{5}$
* Combine $\sqrt{2}$'s: $3\sqrt{2} + \sqrt{2} + \sqrt{2} = 5\sqrt{2}$
* Result: $5\sqrt{2} + 2\sqrt{5}$

33) $\sqrt{45} + \sqrt{24} + \sqrt{24} + \sqrt{20}$
* $\sqrt{45} = 3\sqrt{5}$
* $\sqrt{24} = 2\sqrt{6}$
* $\sqrt{20} = 2\sqrt{5}$
* Combine $\sqrt{5}$'s: $3\sqrt{5} + 2\sqrt{5} = 5\sqrt{5}$
* Combine $\sqrt{6}$'s: $2\sqrt{6} + 2\sqrt{6} = 4\sqrt{6}$
* Result: $5\sqrt{5} + 4\sqrt{6}$

34) $\sqrt{2} + \sqrt{27} + \sqrt{2} + \sqrt{6}$
* $\sqrt{27} = 3\sqrt{3}$
* Combine $\sqrt{2}$'s: $\sqrt{2} + \sqrt{2} = 2\sqrt{2}$
* Result: $2\sqrt{2} + 3\sqrt{3} + \sqrt{6}$

35) $\sqrt{18} + \sqrt{27} + \sqrt{18} + \sqrt{18}$
* $\sqrt{18} = 3\sqrt{2}$
* $\sqrt{27} = 3\sqrt{3}$
* Combine $\sqrt{2}$'s: $3\sqrt{2} + 3\sqrt{2} + 3\sqrt{2} = 9\sqrt{2}$
* Result: $9\sqrt{2} + 3\sqrt{3}$

36) $3\sqrt{20} - 3\sqrt{45} - \sqrt{45}$
* $\sqrt{20} = 2\sqrt{5} \rightarrow 3(2\sqrt{5}) = 6\sqrt{5}$
* $\sqrt{45} = 3\sqrt{5}$
* Expression: $6\sqrt{5} - 3(3\sqrt{5}) - 3\sqrt{5}$
* $6\sqrt{5} - 9\sqrt{5} - 3\sqrt{5} = -6\sqrt{5}$

37) $3\sqrt{5} - \sqrt{20} - 3\sqrt{3}$
* $\sqrt{20} = 2\sqrt{5}$
* $3\sqrt{5} - 2\sqrt{5} - 3\sqrt{3}$
* Combine $\sqrt{5}$'s: $1\sqrt{5} - 3\sqrt{3}$
* Result: $\sqrt{5} - 3\sqrt{3}$

38) $-3\sqrt{5} - \sqrt{18} - 3\sqrt{8}$
* $\sqrt{18} = 3\sqrt{2}$
* $\sqrt{8} = 2\sqrt{2} \rightarrow 3(2\sqrt{2}) = 6\sqrt{2}$
* $-3\sqrt{5} - 3\sqrt{2} - 6\sqrt{2}$
* Combine $\sqrt{2}$'s: $-3\sqrt{2} - 6\sqrt{2} = -9\sqrt{2}$
* Result: $-3\sqrt{5} - 9\sqrt{2}$

39) $-\sqrt{27} - 3\sqrt{6} + 2\sqrt{3}$
* $\sqrt{27} = 3\sqrt{3}$
* $-3\sqrt{3} - 3\sqrt{6} + 2\sqrt{3}$
* Combine $\sqrt{3}$'s: $-3\sqrt{3} + 2\sqrt{3} = -\sqrt{3}$
* Result: $-\sqrt{3} - 3\sqrt{6}$

40) $-2\sqrt{3} - 2\sqrt{12} + 2\sqrt{12}$
* The $-2\sqrt{12}$ and $+2\sqrt{12}$ cancel each other out to 0.
* Result: $-2\sqrt{3}$

41) $-3\sqrt{54} - \sqrt{2} - \sqrt{6} + 2\sqrt{3}$
* $\sqrt{54} = 3\sqrt{6} \rightarrow -3(3\sqrt{6}) = -9\sqrt{6}$
* $-9\sqrt{6} - \sqrt{2} - \sqrt{6} + 2\sqrt{3}$
* Combine $\sqrt{6}$'s: $-9\sqrt{6} - 1\sqrt{6} = -10\sqrt{6}$
* Result: $-10\sqrt{6} - \sqrt{2} + 2\sqrt{3}$

42) $3\sqrt{24} - \sqrt{2} - 3\sqrt{6} + 3\sqrt{54}$
* $\sqrt{24} = 2\sqrt{6} \rightarrow 3(2\sqrt{6}) = 6\sqrt{6}$
* $\sqrt{54} = 3\sqrt{6} \rightarrow 3(3\sqrt{6}) = 9\sqrt{6}$
* $6\sqrt{6} - \sqrt{2} - 3\sqrt{6} + 9\sqrt{6}$
* Combine $\sqrt{6}$'s: $6 - 3 + 9 = 12\sqrt{6}$
* Result: $12\sqrt{6} - \sqrt{2}$

43) $-\sqrt{45} + 3\sqrt{18} - 3\sqrt{20} - \sqrt{24}$
* $\sqrt{45} = 3\sqrt{5}$
* $\sqrt{18} = 3\sqrt{2} \rightarrow 3(3\sqrt{2}) = 9\sqrt{2}$
* $\sqrt{20} = 2\sqrt{5} \rightarrow 3(2\sqrt{5}) = 6\sqrt{5}$
* $\sqrt{24} = 2\sqrt{6}$
* Expression: $-3\sqrt{5} + 9\sqrt{2} - 6\sqrt{5} - 2\sqrt{6}$
* Combine $\sqrt{5}$'s: $-3\sqrt{5} - 6\sqrt{5} = -9\sqrt{5}$
* Result: $-9\sqrt{5} + 9\sqrt{2} - 2\sqrt{6}$

44) $2\sqrt{45} - 3\sqrt{5} - 2\sqrt{45} + 3\sqrt{5}$
* Notice that $2\sqrt{45}$ and $-2\sqrt{45}$ cancel out.
* Notice that $-3\sqrt{5}$ and $+3\sqrt{5}$ cancel out.
* Result: $0$

Final Answer:
A. Simplify
1) $2\sqrt{7}$
2) $6\sqrt{2}$
3) $4\sqrt{2}$
4) $12\sqrt{2}$
5) $6\sqrt{7}$
6) $7\sqrt{2}$
7) $3\sqrt{5}$
8) $8\sqrt{7}$
9) $5\sqrt{6}$
10) $8\sqrt{6}$
11) $-5\sqrt{2}$
12) $24$
13) $-12\sqrt{5}$
14) $18$
15) $25\sqrt{3}$
16) $-32\sqrt{6}$
17) $-7\sqrt{5}$
18) $36$
19) $21\sqrt{2}$
20) $12\sqrt{3}$

B. Simplify (Add/Subt)
21) $4\sqrt{2}$
22) $5\sqrt{5}$
23) $2\sqrt{5}$
24) $4\sqrt{5}$
25) $2\sqrt{2}$
26) $\sqrt{6} + 2\sqrt{2}$
27) $5\sqrt{6}$
28) $6\sqrt{5} + 2\sqrt{6}$
29) $4\sqrt{6} + \sqrt{5}$
30) $3\sqrt{3} + 5\sqrt{2}$
31) $5\sqrt{2} + \sqrt{6} + 3\sqrt{3}$
32) $5\sqrt{2} + 2\sqrt{5}$
33) $5\sqrt{5} + 4\sqrt{6}$
34) $2\sqrt{2} + 3\sqrt{3} + \sqrt{6}$
35) $9\sqrt{2} + 3\sqrt{3}$
36) $-6\sqrt{5}$
37) $\sqrt{5} - 3\sqrt{3}$
38) $-3\sqrt{5} - 9\sqrt{2}$
39) $-\sqrt{3} - 3\sqrt{6}$
40) $-2\sqrt{3}$
41) $-10\sqrt{6} - \sqrt{2} + 2\sqrt{3}$
42) $12\sqrt{6} - \sqrt{2}$
43) $-9\sqrt{5} + 9\sqrt{2} - 2\sqrt{6}$
44) $0$