Squares and Square Roots - Free Printable

To solve the given problems involving square roots of fractions, we need to simplify each fraction inside the square root and then take the square root of both the numerator and the denominator separately. Let's go through each problem step by step.

General Approach:

1. Simplify the fraction inside the square root if possible.
2. Take the square root of the numerator and the denominator separately.
3. Simplify the resulting expression if needed.

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Problem 1: $\sqrt{\frac{10}{40}}$

- Simplify the fraction: $\frac{10}{40} = \frac{1}{4}$.
- Take the square root: $\sqrt{\frac{1}{4}} = \frac{\sqrt{1}}{\sqrt{4}} = \frac{1}{2}$.
- Answer: $\frac{1}{2}$.

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Problem 2: $\sqrt{\frac{11}{99}}$

- Simplify the fraction: $\frac{11}{99} = \frac{1}{9}$.
- Take the square root: $\sqrt{\frac{1}{9}} = \frac{\sqrt{1}}{\sqrt{9}} = \frac{1}{3}$.
- Answer: $\frac{1}{3}$.

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Problem 3: $\sqrt{\frac{25}{49}}$

- The fraction is already simplified.
- Take the square root: $\sqrt{\frac{25}{49}} = \frac{\sqrt{25}}{\sqrt{49}} = \frac{5}{7}$.
- Answer: $\frac{5}{7}$.

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Problem 4: $\sqrt{\frac{10}{640}}$

- Simplify the fraction: $\frac{10}{640} = \frac{1}{64}$.
- Take the square root: $\sqrt{\frac{1}{64}} = \frac{\sqrt{1}}{\sqrt{64}} = \frac{1}{8}$.
- Answer: $\frac{1}{8}$.

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Problem 5: $\sqrt{\frac{200}{128}}$

- Simplify the fraction: $\frac{200}{128} = \frac{25}{16}$.
- Take the square root: $\sqrt{\frac{25}{16}} = \frac{\sqrt{25}}{\sqrt{16}} = \frac{5}{4}$.
- Answer: $\frac{5}{4}$.

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Problem 6: $\sqrt{\frac{324}{441}}$

- The fraction is already simplified.
- Take the square root: $\sqrt{\frac{324}{441}} = \frac{\sqrt{324}}{\sqrt{441}} = \frac{18}{21} = \frac{6}{7}$.
- Answer: $\frac{6}{7}$.

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Problem 7: $\sqrt{\frac{225}{400}}$

- The fraction is already simplified.
- Take the square root: $\sqrt{\frac{225}{400}} = \frac{\sqrt{225}}{\sqrt{400}} = \frac{15}{20} = \frac{3}{4}$.
- Answer: $\frac{3}{4}$.

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Problem 8: $\sqrt{\frac{100}{64}}$

- The fraction is already simplified.
- Take the square root: $\sqrt{\frac{100}{64}} = \frac{\sqrt{100}}{\sqrt{64}} = \frac{10}{8} = \frac{5}{4}$.
- Answer: $\frac{5}{4}$.

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Problem 9: $\sqrt{\frac{192}{27}}$

- Simplify the fraction: $\frac{192}{27} = \frac{64}{9}$.
- Take the square root: $\sqrt{\frac{64}{9}} = \frac{\sqrt{64}}{\sqrt{9}} = \frac{8}{3}$.
- Answer: $\frac{8}{3}$.

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Problem 10: $\sqrt{\frac{18}{32}}$

- Simplify the fraction: $\frac{18}{32} = \frac{9}{16}$.
- Take the square root: $\sqrt{\frac{9}{16}} = \frac{\sqrt{9}}{\sqrt{16}} = \frac{3}{4}$.
- Answer: $\frac{3}{4}$.

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Problem 11: $\sqrt{\frac{20}{80}}$

- Simplify the fraction: $\frac{20}{80} = \frac{1}{4}$.
- Take the square root: $\sqrt{\frac{1}{4}} = \frac{\sqrt{1}}{\sqrt{4}} = \frac{1}{2}$.
- Answer: $\frac{1}{2}$.

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Problem 12: $\sqrt{\frac{4}{64}}$

- Simplify the fraction: $\frac{4}{64} = \frac{1}{16}$.
- Take the square root: $\sqrt{\frac{1}{16}} = \frac{\sqrt{1}}{\sqrt{16}} = \frac{1}{4}$.
- Answer: $\frac{1}{4}$.

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Problem 13: $\sqrt{\frac{54}{96}}$

- Simplify the fraction: $\frac{54}{96} = \frac{9}{16}$.
- Take the square root: $\sqrt{\frac{9}{16}} = \frac{\sqrt{9}}{\sqrt{16}} = \frac{3}{4}$.
- Answer: $\frac{3}{4}$.

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Problem 14: $\sqrt{\frac{27}{75}}$

- Simplify the fraction: $\frac{27}{75} = \frac{9}{25}$.
- Take the square root: $\sqrt{\frac{9}{25}} = \frac{\sqrt{9}}{\sqrt{25}} = \frac{3}{5}$.
- Answer: $\frac{3}{5}$.

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Problem 15: $\sqrt{\frac{10}{90}}$

- Simplify the fraction: $\frac{10}{90} = \frac{1}{9}$.
- Take the square root: $\sqrt{\frac{1}{9}} = \frac{\sqrt{1}}{\sqrt{9}} = \frac{1}{3}$.
- Answer: $\frac{1}{3}$.

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Problem 16: $\sqrt{\frac{20}{125}}$

- Simplify the fraction: $\frac{20}{125} = \frac{4}{25}$.
- Take the square root: $\sqrt{\frac{4}{25}} = \frac{\sqrt{4}}{\sqrt{25}} = \frac{2}{5}$.
- Answer: $\frac{2}{5}$.

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Problem 17: $\sqrt{\frac{200}{800}}$

- Simplify the fraction: $\frac{200}{800} = \frac{1}{4}$.
- Take the square root: $\sqrt{\frac{1}{4}} = \frac{\sqrt{1}}{\sqrt{4}} = \frac{1}{2}$.
- Answer: $\frac{1}{2}$.

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Problem 18: $\sqrt{\frac{110}{990}}$

- Simplify the fraction: $\frac{110}{990} = \frac{1}{9}$.
- Take the square root: $\sqrt{\frac{1}{9}} = \frac{\sqrt{1}}{\sqrt{9}} = \frac{1}{3}$.
- Answer: $\frac{1}{3}$.

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Problem 19: $\sqrt{\frac{125}{500}}$

- Simplify the fraction: $\frac{125}{500} = \frac{1}{4}$.
- Take the square root: $\sqrt{\frac{1}{4}} = \frac{\sqrt{1}}{\sqrt{4}} = \frac{1}{2}$.
- Answer: $\frac{1}{2}$.

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Problem 20: $\sqrt{\frac{10}{250}}$

- Simplify the fraction: $\frac{10}{250} = \frac{1}{25}$.
- Take the square root: $\sqrt{\frac{1}{25}} = \frac{\sqrt{1}}{\sqrt{25}} = \frac{1}{5}$.
- Answer: $\frac{1}{5}$.

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Problem 21: $\sqrt{\frac{100}{625}}$

- Simplify the fraction: $\frac{100}{625} = \frac{4}{25}$.
- Take the square root: $\sqrt{\frac{4}{25}} = \frac{\sqrt{4}}{\sqrt{25}} = \frac{2}{5}$.
- Answer: $\frac{2}{5}$.

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Problem 22: $\sqrt{\frac{12}{48}}$

- Simplify the fraction: $\frac{12}{48} = \frac{1}{4}$.
- Take the square root: $\sqrt{\frac{1}{4}} = \frac{\sqrt{1}}{\sqrt{4}} = \frac{1}{2}$.
- Answer: $\frac{1}{2}$.

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Problem 23: $\sqrt{\frac{2}{72}}$

- Simplify the fraction: $\frac{2}{72} = \frac{1}{36}$.
- Take the square root: $\sqrt{\frac{1}{36}} = \frac{\sqrt{1}}{\sqrt{36}} = \frac{1}{6}$.
- Answer: $\frac{1}{6}$.

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Problem 24: $\sqrt{\frac{3}{75}}$

- Simplify the fraction: $\frac{3}{75} = \frac{1}{25}$.
- Take the square root: $\sqrt{\frac{1}{25}} = \frac{\sqrt{1}}{\sqrt{25}} = \frac{1}{5}$.
- Answer: $\frac{1}{5}$.

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Final Answer:

\[
\boxed{
\begin{array}{ccc}
\frac{1}{2} & \frac{1}{3} & \frac{5}{7} \\
\frac{1}{8} & \frac{5}{4} & \frac{6}{7} \\
\frac{3}{4} & \frac{5}{4} & \frac{8}{3} \\
\frac{3}{4} & \frac{1}{2} & \frac{1}{4} \\
\frac{3}{4} & \frac{3}{5} & \frac{1}{3} \\
\frac{2}{5} & \frac{1}{2} & \frac{1}{3} \\
\frac{1}{2} & \frac{1}{5} & \frac{2}{5} \\
\frac{1}{2} & \frac{1}{6} & \frac{1}{5} \\
\end{array}
}
\]