Simplifying Radicals worksheet with exercises to complete tables and express radicals in simplified form.

To solve this worksheet on simplifying radicals, we need to break down each square root into a product of a perfect square and another number. Then, we take the square root of the perfect square and move it outside the radical sign.

Here is the step-by-step completion for all sections:

Section A: Completing the Tables

Table 1 (Multiples of 2):
* $\sqrt{8}$: Break into $\sqrt{4 \times 2}$. Since $\sqrt{4}=2$, answer is $2\sqrt{2}$.
* $\sqrt{18}$: Break into $\sqrt{9 \times 2}$. Since $\sqrt{9}=3$, answer is $3\sqrt{2}$.
* $\sqrt{72}$: Break into $\sqrt{36 \times 2}$. Since $\sqrt{36}=6$, answer is $6\sqrt{2}$.
* $\sqrt{200}$: Break into $\sqrt{100 \times 2}$. Since $\sqrt{100}=10$, answer is $10\sqrt{2}$.

Table 2 (Multiples of 5):
* $\sqrt{80}$: Break into $\sqrt{16 \times 5}$. Since $\sqrt{16}=4$, answer is $4\sqrt{5}$.
* $\sqrt{180}$: Break into $\sqrt{36 \times 5}$. Since $\sqrt{36}=6$, answer is $6\sqrt{5}$.
* Row with $\sqrt{100 \times 5}$: The original radical is $\sqrt{500}$. The simplified form is $10\sqrt{5}$.
* Row with $9\sqrt{5}$: The intermediate step is $\sqrt{81 \times 5}$, so the original radical is $\sqrt{405}$.
* Row with $11\sqrt{5}$: The intermediate step is $\sqrt{121 \times 5}$, so the original radical is $\sqrt{605}$.

Table 3 (Multiples of 3):
* $\sqrt{12}$: Break into $\sqrt{4 \times 3}$. Since $\sqrt{4}=2$, answer is $2\sqrt{3}$.
* $\sqrt{75}$: Break into $\sqrt{25 \times 3}$. Since $\sqrt{25}=5$, answer is $5\sqrt{3}$.
* $\sqrt{48}$: Break into $\sqrt{16 \times 3}$. Since $\sqrt{16}=4$, answer is $4\sqrt{3}$.
* Row with $\sqrt{36 \times 3}$: The original radical is $\sqrt{108}$. The simplified form is $6\sqrt{3}$.
* Row with $\sqrt{144 \times 3}$: The original radical is $\sqrt{432}$. The simplified form is $12\sqrt{3}$.

Table 4 (Multiples of 7):
* $\sqrt{63}$: Break into $\sqrt{9 \times 7}$. Since $\sqrt{9}=3$, answer is $3\sqrt{7}$.
* $\sqrt{252}$: Break into $\sqrt{36 \times 7}$. Since $\sqrt{36}=6$, answer is $6\sqrt{7}$.
* Row with $\sqrt{81 \times 7}$: The original radical is $\sqrt{567}$. The simplified form is $9\sqrt{7}$.
* Row with $7\sqrt{7}$: The intermediate step is $\sqrt{49 \times 7}$, so the original radical is $\sqrt{343}$.
* Row with $12\sqrt{7}$: The intermediate step is $\sqrt{144 \times 7}$, so the original radical is $\sqrt{1008}$.

Section B: Writing in simplest form ($a\sqrt{b}$)

1) $\sqrt{32}$: $\sqrt{16 \times 2} = 4\sqrt{2}$
2) $\sqrt{27}$: $\sqrt{9 \times 3} = 3\sqrt{3}$
3) $\sqrt{20}$: $\sqrt{4 \times 5} = 2\sqrt{5}$
4) $\sqrt{300}$: $\sqrt{100 \times 3} = 10\sqrt{3}$
5) $\sqrt{45}$: $\sqrt{9 \times 5} = 3\sqrt{5}$
6) $\sqrt{98}$: $\sqrt{49 \times 2} = 7\sqrt{2}$
7) $\sqrt{28}$: $\sqrt{4 \times 7} = 2\sqrt{7}$
8) $\sqrt{44}$: $\sqrt{4 \times 11} = 2\sqrt{11}$
9) $\sqrt{128}$: $\sqrt{64 \times 2} = 8\sqrt{2}$
10) $\sqrt{320}$: $\sqrt{64 \times 5} = 8\sqrt{5}$
11) $\sqrt{162}$: $\sqrt{81 \times 2} = 9\sqrt{2}$
12) $\sqrt{147}$: $\sqrt{49 \times 3} = 7\sqrt{3}$
13) $\sqrt{24}$: $\sqrt{4 \times 6} = 2\sqrt{6}$
14) $\sqrt{810}$: $\sqrt{81 \times 10} = 9\sqrt{10}$
15) $\sqrt{60}$: $\sqrt{4 \times 15} = 2\sqrt{15}$
(Note: There are two questions labeled 15 in the image. The second one is below question 14).
15) $\sqrt{56}$: $\sqrt{4 \times 14} = 2\sqrt{14}$

Final Answer:
Section A Answers:

Top Left Table:
- $\sqrt{8} \rightarrow \sqrt{4 \times 2} \rightarrow 2\sqrt{2}$
- $\sqrt{18} \rightarrow \sqrt{9 \times 2} \rightarrow 3\sqrt{2}$
- $\sqrt{72} \rightarrow \sqrt{36 \times 2} \rightarrow 6\sqrt{2}$
- $\sqrt{200} \rightarrow \sqrt{100 \times 2} \rightarrow 10\sqrt{2}$

Top Right Table:
- $\sqrt{80} \rightarrow \sqrt{16 \times 5} \rightarrow 4\sqrt{5}$
- $\sqrt{180} \rightarrow \sqrt{36 \times 5} \rightarrow 6\sqrt{5}$
- Row 3: $\sqrt{500} \rightarrow \sqrt{100 \times 5} \rightarrow 10\sqrt{5}$
- Row 4: $\sqrt{405} \rightarrow \sqrt{81 \times 5} \rightarrow 9\sqrt{5}$
- Row 5: $\sqrt{605} \rightarrow \sqrt{121 \times 5} \rightarrow 11\sqrt{5}$

Bottom Left Table:
- $\sqrt{12} \rightarrow \sqrt{4 \times 3} \rightarrow 2\sqrt{3}$
- $\sqrt{75} \rightarrow \sqrt{25 \times 3} \rightarrow 5\sqrt{3}$
- $\sqrt{48} \rightarrow \sqrt{16 \times 3} \rightarrow 4\sqrt{3}$
- Row 4: $\sqrt{108} \rightarrow \sqrt{36 \times 3} \rightarrow 6\sqrt{3}$
- Row 5: $\sqrt{432} \rightarrow \sqrt{144 \times 3} \rightarrow 12\sqrt{3}$

Bottom Right Table:
- $\sqrt{63} \rightarrow \sqrt{9 \times 7} \rightarrow 3\sqrt{7}$
- $\sqrt{252} \rightarrow \sqrt{36 \times 7} \rightarrow 6\sqrt{7}$
- Row 3: $\sqrt{567} \rightarrow \sqrt{81 \times 7} \rightarrow 9\sqrt{7}$
- Row 4: $\sqrt{343} \rightarrow \sqrt{49 \times 7} \rightarrow 7\sqrt{7}$
- Row 5: $\sqrt{1008} \rightarrow \sqrt{144 \times 7} \rightarrow 12\sqrt{7}$

Section B Answers:
1) $4\sqrt{2}$
2) $3\sqrt{3}$
3) $2\sqrt{5}$
4) $10\sqrt{3}$
5) $3\sqrt{5}$
6) $7\sqrt{2}$
7) $2\sqrt{7}$
8) $2\sqrt{11}$
9) $8\sqrt{2}$
10) $8\sqrt{5}$
11) $9\sqrt{2}$
12) $7\sqrt{3}$
13) $2\sqrt{6}$
14) $9\sqrt{10}$
15) $2\sqrt{15}$
15) $2\sqrt{14}$