Principal Roots Worksheet - Free Printable

Let's solve the worksheet step by step.

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Activity: Principal Roots

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## Part A: Determine the principal roots in each item.
> If the root is irrational, write IRR.
> If the root is imaginary, write IM.

We'll evaluate each square or cube root:

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1) $ \sqrt{49} = 7 $
✔ Perfect square → 7

2) $ \sqrt{8} $
8 is not a perfect square.
$ \sqrt{8} = \sqrt{4 \cdot 2} = 2\sqrt{2} $ → irrational
→ IRR

3) $ \sqrt{64} = 8 $
Perfect square → 8

4) $ \sqrt{81} = 9 $
Perfect square → 9

5) $ \sqrt{-9} $
Negative number under square root → imaginary
→ IM

6) $ \sqrt{196} = 14 $
Because $ 14^2 = 196 $ → 14

7) $ \sqrt{-36} $
Negative under square root → imaginary
→ IM

8) $ \sqrt{27} $
Not a perfect square: $ \sqrt{27} = \sqrt{9 \cdot 3} = 3\sqrt{3} $ → irrational
→ IRR

9) $ \sqrt[3]{64} $
Cube root of 64: $ 4^3 = 64 $ → 4

10) $ \sqrt[3]{-8} $
Cube root of -8: $ (-2)^3 = -8 $ → -2

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✔ Part A Answers:

| Item | Answer |
|------|--------|
| 1) | 7 |
| 2) | IRR |
| 3) | 8 |
| 4) | 9 |
| 5) | IM |
| 6) | 14 |
| 7) | IM |
| 8) | IRR |
| 9) | 4 |
| 10) | -2 |

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## Part B: Determine which two integers the principal root lies between.

We need to find two consecutive integers such that the root is between them.

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1) $ \sqrt{40} $
Find perfect squares near 40:
$ 6^2 = 36 $, $ 7^2 = 49 $
So: $ 6 < \sqrt{40} < 7 $

→ 6 < √40 < 7

2) $ \sqrt{8} $
$ 2^2 = 4 $, $ 3^2 = 9 $
So: $ 2 < \sqrt{8} < 3 $

→ 2 < √8 < 3

3) $ \sqrt{125} $
$ 10^2 = 100 $, $ 11^2 = 121 $, $ 12^2 = 144 $
125 is between 121 and 144 → $ 11^2 = 121 $, $ 12^2 = 144 $
So: $ 11 < \sqrt{125} < 12 $

→ 11 < √125 < 12

4) $ \sqrt{240} $
$ 15^2 = 225 $, $ 16^2 = 256 $
240 is between 225 and 256 → $ 15 < \sqrt{240} < 16 $

→ 15 < √240 < 16

5) $ \sqrt[3]{36} $
Cube roots:
$ 3^3 = 27 $, $ 4^3 = 64 $
So: $ 3 < \sqrt[3]{36} < 4 $

→ 3 < ∛36 < 4

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✔ Part B Answers:

| Item | Answer |
|------|--------|
| 1) | 6 < √40 < 7 |
| 2) | 2 < √8 < 3 |
| 3) | 11 < √125 < 12 |
| 4) | 15 < √240 < 16 |
| 5) | 3 < ∛36 < 4 |

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✔ Final Answer Summary:

#### Part A:
1) 7
2) IRR
3) 8
4) 9
5) IM
6) 14
7) IM
8) IRR
9) 4
10) -2

#### Part B:
1) 6 < √40 < 7
2) 2 < √8 < 3
3) 11 < √125 < 12
4) 15 < √240 < 16
5) 3 < ∛36 < 4

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