Estimating Values of Square and Cube Roots Worksheet | 8th Grade ... - Free Printable

Let's solve each section of this worksheet step by step and explain the reasoning behind each answer.

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Section A: Estimate using number lines

We are to estimate square roots based on nearby perfect squares.

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#### 1) Estimate √75

- The number line shows:
- $8^2 = 64$
- $9^2 = 81$

So, $ \sqrt{75} $ is between 8 and 9.

Now, how close is 75 to 64 and 81?
- Distance from 64 to 75: $75 - 64 = 11$
- Distance from 75 to 81: $81 - 75 = 6$

Since 75 is closer to 81 than to 64, $ \sqrt{75} $ is closer to 9 than to 8.

Approximate value:
$$
\sqrt{75} \approx 8.7
$$

✔ Answer: $ \boxed{8.7} $

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#### 2) Estimate √110

- $10^2 = 100$
- $11^2 = 121$

So, $ \sqrt{110} $ is between 10 and 11.

- $110 - 100 = 10$
- $121 - 110 = 11$

So it's slightly closer to 10 than to 11.

Approximate:
$$
\sqrt{110} \approx 10.5
$$

✔ Answer: $ \boxed{10.5} $

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#### 3) Estimate √39

- $6^2 = 36$
- We need $7^2 = 49$ (not shown, but implied)

So $ \sqrt{39} $ is between 6 and 7.

- $39 - 36 = 3$
- $49 - 39 = 10$

Closer to 6.

Approximate:
$$
\sqrt{39} \approx 6.2
$$

✔ Answer: $ \boxed{6.2} $

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Section B: Complete the table

We have cube roots and powers to match.

| Cube Root | Value |
|----------|-------|
| $ \sqrt[3]{1} $ | $1^3 = 1$ → $ \boxed{1} $ |
| $ \sqrt[3]{8} $ | $2^3 = 8$ → $ \boxed{2} $ |
| $ \sqrt[3]{27} $ | $3^3 = 27$ → $ \boxed{3} $ |
| $ \sqrt[3]{64} $ | $4^3 = 64$ → $ \boxed{4} $ |
| $ \sqrt[3]{125} $ | $5^3 = 125$ → $ \boxed{5} $ |
| $ \sqrt[3]{216} $ | $6^3 = 216$ → $ \boxed{6} $ |
| $ \sqrt[3]{343} $ | $7^3 = 343$ → $ \boxed{7} $ |
| $ \sqrt[3]{1000} $ | $10^3 = 1000$ → $ \boxed{10} $ |
| $ \sqrt[3]{729} $ | $9^3 = 729$ → $ \boxed{9} $ |
| $ \sqrt[3]{512} $ | $8^3 = 512$ → $ \boxed{8} $ |

✔ Completed Table:

| $ \sqrt[3]{1} $ | $ \sqrt[3]{8} $ | $ \sqrt[3]{27} $ | $ \sqrt[3]{64} $ | $ \sqrt[3]{125} $ | $ \sqrt[3]{216} $ | $ \sqrt[3]{343} $ | $ \sqrt[3]{1000} $ | $ \sqrt[3]{729} $ | $ \sqrt[3]{512} $ |
|------------------|------------------|-------------------|-------------------|---------------------|----------------------|----------------------|------------------------|----------------------|----------------------|
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 10 | 9 | 8 |

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Section C: Approximate square or cube roots

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#### Left Table: Square Roots → Interval

We find between which two consecutive integers the square root lies.

1. $ \sqrt{6} $
- $2^2 = 4$, $3^2 = 9$
- So $ \sqrt{6} $ is between 2 and 3.
✔ Interval: $ \boxed{2 \sim 3} $

2. $ \sqrt{15} $
- $3^2 = 9$, $4^2 = 16$
- So $ \sqrt{15} $ is between 3 and 4.
✔ Interval: $ \boxed{3 \sim 4} $

3. $ \sqrt{27} $
- $5^2 = 25$, $6^2 = 36$
- So $ \sqrt{27} $ is between 5 and 6.
✔ Interval: $ \boxed{5 \sim 6} $

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#### Right Table: Square Root → Given Interval → Find Expression

We reverse: given interval, find a square root that lies in it.

1. Interval: $11 \sim 12$
- So we want $ \sqrt{x} $ such that $11 < \sqrt{x} < 12$
- Square both ends: $121 < x < 144$
- Pick any number in that range, e.g., $x = 130$
- So $ \sqrt{130} $ is between 11 and 12
✔ Example: $ \boxed{\sqrt{130}} $

2. Interval: $0.8 \sim 0.9$
- $0.8^2 = 0.64$, $0.9^2 = 0.81$
- So pick a number between 0.64 and 0.81, say 0.75
- $ \sqrt{0.75} \approx 0.866 $
✔ Example: $ \boxed{\sqrt{0.75}} $

3. Interval: $0.15 \sim 0.16$
- $0.15^2 = 0.0225$, $0.16^2 = 0.0256$
- Pick $x = 0.024$
- $ \sqrt{0.024} \approx 0.155 $
✔ Example: $ \boxed{\sqrt{0.024}} $

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#### Cube Roots: Left Table

1. $ \sqrt[3]{21} $
- $2^3 = 8$, $3^3 = 27$
- So $ \sqrt[3]{21} $ is between 2 and 3
✔ Interval: $ \boxed{2 \sim 3} $

2. $ \sqrt[3]{200} $
- $5^3 = 125$, $6^3 = 216$
- So $ \sqrt[3]{200} $ is between 5 and 6
✔ Interval: $ \boxed{5 \sim 6} $

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#### Cube Roots: Right Table

Given interval → find cube root expression

1. Interval: $4 \sim 5$
- $4^3 = 64$, $5^3 = 125$
- Pick $x = 100$, $ \sqrt[3]{100} \approx 4.64 $
✔ Example: $ \boxed{\sqrt[3]{100}} $

2. Interval: $9 \sim 10$
- $9^3 = 729$, $10^3 = 1000$
- Pick $x = 800$, $ \sqrt[3]{800} \approx 9.28 $
✔ Example: $ \boxed{\sqrt[3]{800}} $

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✔ Final Answers Summary:

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#### Section A:
1. $ \sqrt{75} \approx \boxed{8.7} $
2. $ \sqrt{110} \approx \boxed{10.5} $
3. $ \sqrt{39} \approx \boxed{6.2} $

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#### Section B:
| $ \sqrt[3]{1} $ | $ \sqrt[3]{8} $ | $ \sqrt[3]{27} $ | $ \sqrt[3]{64} $ | $ \sqrt[3]{125} $ | $ \sqrt[3]{216} $ | $ \sqrt[3]{343} $ | $ \sqrt[3]{1000} $ | $ \sqrt[3]{729} $ | $ \sqrt[3]{512} $ |
|------------------|------------------|-------------------|-------------------|---------------------|----------------------|----------------------|------------------------|----------------------|----------------------|
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 10 | 9 | 8 |

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#### Section C:

##### Square Roots → Interval:
- $ \sqrt{6} $: $ \boxed{2 \sim 3} $
- $ \sqrt{15} $: $ \boxed{3 \sim 4} $
- $ \sqrt{27} $: $ \boxed{5 \sim 6} $

##### Square Root → Given Interval:
- $11 \sim 12$: $ \boxed{\sqrt{130}} $
- $0.8 \sim 0.9$: $ \boxed{\sqrt{0.75}} $
- $0.15 \sim 0.16$: $ \boxed{\sqrt{0.024}} $

##### Cube Roots → Interval:
- $ \sqrt[3]{21} $: $ \boxed{2 \sim 3} $
- $ \sqrt[3]{200} $: $ \boxed{5 \sim 6} $

##### Cube Root → Given Interval:
- $4 \sim 5$: $ \boxed{\sqrt[3]{100}} $
- $9 \sim 10$: $ \boxed{\sqrt[3]{800}} $

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