Square Roots Worksheet with 12 math problems involving square roots, multiplication, addition, subtraction, and division.

Let's solve each problem on the Square Roots Worksheet step by step and explain the reasoning.

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1a. $\sqrt{49} \times \sqrt{49}$

- $\sqrt{49} = 7$, because $7 \times 7 = 49$
- So, $7 \times 7 = 49$

✔ Answer: 49

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1b. $(\sqrt{16})^2$

- $\sqrt{16} = 4$, because $4 \times 4 = 16$
- Then, $4^2 = 16$

✔ Answer: 16

> *Note:* Squaring a square root cancels out the square root: $(\sqrt{a})^2 = a$

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2a. $\sqrt{2 \times 8}$

- First, simplify inside the square root: $2 \times 8 = 16$
- $\sqrt{16} = 4$

✔ Answer: 4

> *Alternative way:* $\sqrt{2 \times 8} = \sqrt{16} = 4$

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2b. $\sqrt{36} - \sqrt{81}$

- $\sqrt{36} = 6$
- $\sqrt{81} = 9$
- $6 - 9 = -3$

✔ Answer: -3

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3a. $\sqrt{91 - 27}$

- First, subtract: $91 - 27 = 64$
- $\sqrt{64} = 8$

✔ Answer: 8

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3b. $\sqrt{\frac{490}{10}}$

- Simplify inside: $\frac{490}{10} = 49$
- $\sqrt{49} = 7$

✔ Answer: 7

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4a. $\sqrt{24^2}$

- $24^2 = 576$
- $\sqrt{576} = 24$

✔ Answer: 24

> *Note:* $\sqrt{a^2} = |a|$, but since $24 > 0$, it's just $24$

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4b. $\sqrt{49} + \sqrt{100}$

- $\sqrt{49} = 7$
- $\sqrt{100} = 10$
- $7 + 10 = 17$

✔ Answer: 17

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5a. $\sqrt{49 + 0}$

- $49 + 0 = 49$
- $\sqrt{49} = 7$

✔ Answer: 7

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5b. $\sqrt{9} \times \sqrt{49}$

- $\sqrt{9} = 3$
- $\sqrt{49} = 7$
- $3 \times 7 = 21$

✔ Answer: 21

> *Property:* $\sqrt{a} \times \sqrt{b} = \sqrt{a \times b}$
> So, $\sqrt{9} \times \sqrt{49} = \sqrt{9 \times 49} = \sqrt{441} = 21$

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6a. $\frac{\sqrt{100}}{\sqrt{4}}$

- $\sqrt{100} = 10$
- $\sqrt{4} = 2$
- $\frac{10}{2} = 5$

✔ Answer: 5

> *Property:* $\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$
> So, $\frac{\sqrt{100}}{\sqrt{4}} = \sqrt{\frac{100}{4}} = \sqrt{25} = 5$

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6b. $(\sqrt{100})^2$

- $\sqrt{100} = 10$
- $10^2 = 100$

✔ Answer: 100

> Again, squaring a square root cancels it: $(\sqrt{a})^2 = a$

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✔ Final Answers:

| Problem | Answer |
|--------|--------|
| 1a | 49 |
| 1b | 16 |
| 2a | 4 |
| 2b | -3 |
| 3a | 8 |
| 3b | 7 |
| 4a | 24 |
| 4b | 17 |
| 5a | 7 |
| 5b | 21 |
| 6a | 5 |
| 6b | 100 |

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Let me know if you'd like this formatted as a printable answer key!