Solving with Squared and Cubed Worksheet Download - Free Printable

The image you've uploaded is a math worksheet titled *"Solving with Squared and Cubed"*, designed to help students practice solving equations involving squares and cubes. The goal is to find the positive value of $ x $ for each equation.

Let’s go through the problems one by one, explain how to solve them, and verify the answers provided in the "Answer Key."

---

🔹 General Rules:

- For $ x^2 = a $:
$ x = \sqrt{a} $ (we take the positive square root, as per instruction)
- For $ x^3 = a $:
$ x = \sqrt[3]{a} $ (cube root — this can be positive or negative, but we're asked for positive values)

---

✔ Problem-by-Problem Solution:

#### 1) $ x^2 = 4 $
$$
x = \sqrt{4} = 2
$$
✔ Answer: 2

#### 2) $ x^2 = 9 $
$$
x = \sqrt{9} = 3
$$
✔ Answer: 3

#### 3) $ x^3 = 125 $
$$
x = \sqrt[3]{125} = 5 \quad (\text{since } 5^3 = 125)
$$
✔ Answer: 5

#### 4) $ x^2 = 144 $
$$
x = \sqrt{144} = 12
$$
✔ Answer: 12

#### 5) $ x^2 = 49 $
$$
x = \sqrt{49} = 7
$$
✔ Answer: 7

#### 6) $ x^3 = 64 $
$$
x = \sqrt[3]{64} = 4 \quad (\text{since } 4^3 = 64)
$$
✔ Answer: 4

#### 7) $ x^2 = 81 $
$$
x = \sqrt{81} = 9
$$
✔ Answer: 9

#### 8) $ x^2 = 25 $
$$
x = \sqrt{25} = 5
$$
✔ Answer: 5

#### 9) $ x^3 = 1 $
$$
x = \sqrt[3]{1} = 1
$$
✔ Answer: 1

#### 10) $ x^3 = 1,000 $
$$
x = \sqrt[3]{1000} = 10 \quad (\text{since } 10^3 = 1000)
$$
✔ Answer: 10

#### 11) $ x^3 = 8 $
$$
x = \sqrt[3]{8} = 2 \quad (\text{since } 2^3 = 8)
$$
✔ Answer: 2

#### 12) $ x^3 = 512 $
$$
x = \sqrt[3]{512} = 8 \quad (\text{since } 8^3 = 512)
$$
✔ Answer: 8

#### 13) $ x^2 = 64 $
$$
x = \sqrt{64} = 8
$$
✔ Answer: 8

#### 14) $ x^3 = 343 $
$$
x = \sqrt[3]{343} = 7 \quad (\text{since } 7^3 = 343)
$$
✔ Answer: 7

#### 15) $ x^3 = 216 $
$$
x = \sqrt[3]{216} = 6 \quad (\text{since } 6^3 = 216)
$$
✔ Answer: 6

#### 16) $ x^2 = 121 $
$$
x = \sqrt{121} = 11
$$
✔ Answer: 11

#### 17) $ x^2 = 100 $
$$
x = \sqrt{100} = 10
$$
✔ Answer: 10

#### 18) $ x^3 = 729 $
$$
x = \sqrt[3]{729} = 9 \quad (\text{since } 9^3 = 729)
$$
✔ Answer: 9

#### 19) $ x^2 = 36 $
$$
x = \sqrt{36} = 6
$$
✔ Answer: 6

#### 20) $ x^2 = 16 $
$$
x = \sqrt{16} = 4
$$
✔ Answer: 4

#### 21) $ x^3 = 27 $
$$
x = \sqrt[3]{27} = 3 \quad (\text{since } 3^3 = 27)
$$
✔ Answer: 3

---

✔ Final Answer Key:

| Problem | Answer |
|--------|--------|
| 1 | 2 |
| 2 | 3 |
| 3 | 5 |
| 4 | 12 |
| 5 | 7 |
| 6 | 4 |
| 7 | 9 |
| 8 | 5 |
| 9 | 1 |
| 10 | 10 |
| 11 | 2 |
| 12 | 8 |
| 13 | 8 |
| 14 | 7 |
| 15 | 6 |
| 16 | 11 |
| 17 | 10 |
| 18 | 9 |
| 19 | 6 |
| 20 | 4 |
| 21 | 3 |

All answers match the provided answer key.

---

📌 Summary:

This worksheet focuses on:
- Finding square roots for $ x^2 = a $
- Finding cube roots for $ x^3 = a $
- Always choosing the positive solution (as specified)

Each problem uses perfect squares and cubes, so no approximations are needed.

✔ All solutions are correct and consistent with the answer key.

Let me know if you'd like a printable version or explanation in another format!