Math practice worksheet focusing on cube roots and number line placement.

Problem Analysis:

The task involves filling in the blanks to determine the cube root of each number provided. The goal is to work backward from the given cube roots to find the original numbers. Additionally, there is a practice section where you need to approximate the values of square roots and cube roots on a number line.

---

Step-by-Step Solution:

#### Part 1: Determine the Cube Roots

We are given several cube roots and need to fill in the blanks for the original numbers. Let's solve each one step by step.

1. Given: $ \sqrt[3]{x} = 27 $
- To find $ x $, we cube both sides:
$$
x = 27^3
$$
Calculate $ 27^3 $:
$$
27^3 = 27 \times 27 \times 27 = 729 \times 27 = 19683
$$
Therefore, $ x = 19683 $.

2. Given: $ \sqrt[3]{x} = 12 $
- To find $ x $, we cube both sides:
$$
x = 12^3
$$
Calculate $ 12^3 $:
$$
12^3 = 12 \times 12 \times 12 = 144 \times 12 = 1728
$$
Therefore, $ x = 1728 $.

3. Given: $ \sqrt[3]{x} = 1 $
- To find $ x $, we cube both sides:
$$
x = 1^3
$$
Calculate $ 1^3 $:
$$
1^3 = 1
$$
Therefore, $ x = 1 $.

4. Given: $ \sqrt[3]{x} = 8 $
- To find $ x $, we cube both sides:
$$
x = 8^3
$$
Calculate $ 8^3 $:
$$
8^3 = 8 \times 8 \times 8 = 64 \times 8 = 512
$$
Therefore, $ x = 512 $.

5. Given: $ \sqrt[3]{x} = 0.8 $
- To find $ x $, we cube both sides:
$$
x = 0.8^3
$$
Calculate $ 0.8^3 $:
$$
0.8^3 = 0.8 \times 0.8 \times 0.8 = 0.64 \times 0.8 = 0.512
$$
Therefore, $ x = 0.512 $.

---

#### Part 2: Practice Section

In this section, you are asked to approximate the values of square roots and cube roots on a number line. Let's approximate each value:

1. $ \sqrt{2} $:
- The square root of 2 is approximately 1.414.
- On the number line, place it between 1 and 2, closer to 1.4.

2. $ \sqrt{3} $:
- The square root of 3 is approximately 1.732.
- On the number line, place it between 1 and 2, closer to 1.7.

3. $ \sqrt{5} $:
- The square root of 5 is approximately 2.236.
- On the number line, place it between 2 and 3, closer to 2.2.

4. $ \sqrt[3]{3} $:
- The cube root of 3 is approximately 1.442.
- On the number line, place it between 1 and 2, closer to 1.4.

5. $ \sqrt[3]{5} $:
- The cube root of 5 is approximately 1.710.
- On the number line, place it between 1 and 2, closer to 1.7.

6. $ \sqrt[3]{7} $:
- The cube root of 7 is approximately 1.913.
- On the number line, place it between 1 and 2, closer to 1.9.

7. $ \sqrt[3]{10} $:
- The cube root of 10 is approximately 2.154.
- On the number line, place it between 2 and 3, closer to 2.2.

8. $ \sqrt[3]{12} $:
- The cube root of 12 is approximately 2.289.
- On the number line, place it between 2 and 3, closer to 2.3.

---

Final Answers:

#### Part 1:
1. $ x = 19683 $
2. $ x = 1728 $
3. $ x = 1 $
4. $ x = 512 $
5. $ x = 0.512 $

#### Part 2:
- $ \sqrt{2} \approx 1.414 $
- $ \sqrt{3} \approx 1.732 $
- $ \sqrt{5} \approx 2.236 $
- $ \sqrt[3]{3} \approx 1.442 $
- $ \sqrt[3]{5} \approx 1.710 $
- $ \sqrt[3]{7} \approx 1.913 $
- $ \sqrt[3]{10} \approx 2.154 $
- $ \sqrt[3]{12} \approx 2.289 $

---

Boxed Final Answer:

$$
\boxed{
\begin{aligned}
&\text{Part 1:} \\
&1. \, x = 19683 \\
&2. \, x = 1728 \\
&3. \, x = 1 \\
&4. \, x = 512 \\
&5. \, x = 0.512 \\
\\
&\text{Part 2:} \\
&\sqrt{2} \approx 1.414, \, \sqrt{3} \approx 1.732, \, \sqrt{5} \approx 2.236, \\
&\sqrt[3]{3} \approx 1.442, \, \sqrt[3]{5} \approx 1.710, \, \sqrt[3]{7} \approx 1.913, \\
&\sqrt[3]{10} \approx 2.154, \, \sqrt[3]{12} \approx 2.289
\end{aligned}
}
$$