Math worksheet focusing on squares and square roots with exercises to identify perfect squares, calculate square roots, and simplify expressions.

Let's solve each part of the worksheet step by step and explain the reasoning.

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Part 1: Determine whether the following numbers are perfect squares.

A perfect square is a number that can be expressed as the square of an integer (e.g., $ 16 = 4^2 $).

#### a) 120
- Check: Is there an integer whose square is 120?
$ 10^2 = 100 $, $ 11^2 = 121 $ → 120 is between them.
→ Not a perfect square.
✔ NO

#### b) 144
- $ 12^2 = 144 $ → Yes!
✔ YES

#### c) 160
- $ 12^2 = 144 $, $ 13^2 = 169 $ → 160 is in between.
→ Not a perfect square.
✔ NO

#### d) 196
- $ 14^2 = 196 $ → Yes!
✔ YES

#### e) 250
- $ 15^2 = 225 $, $ 16^2 = 256 $ → 250 is between them.
→ Not a perfect square.
✔ NO

#### f) 300
- $ 17^2 = 289 $, $ 18^2 = 324 $ → 300 is between them.
→ Not a perfect square.
✔ NO

#### g) 324
- $ 18^2 = 324 $ → Yes!
✔ YES

#### h) 484
- $ 22^2 = 484 $ → Yes!
✔ YES

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✔ Answers for Part 1:

a) NO
b) YES
c) NO
d) YES
e) NO
f) NO
g) YES
h) YES

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Part 2: Determine the square roots of the following.

#### a) $ \sqrt{36} $
- $ 6^2 = 36 $ → $ \sqrt{36} = 6 $
✔ 6

#### b) $ \sqrt{7^2} $
- $ 7^2 = 49 $, so $ \sqrt{7^2} = 7 $
(Not 49 or 14 — remember, square root of a square returns the original number if positive.)
✔ 7

#### c) $ \sqrt{0.64} $
- Think: What number squared gives 0.64?
$ 0.8^2 = 0.64 $ → So $ \sqrt{0.64} = 0.8 $
✔ 0.8

#### d) $ \sqrt{\frac{9}{25}} $
- $ \frac{9}{25} = \left(\frac{3}{5}\right)^2 $ → So $ \sqrt{\frac{9}{25}} = \frac{3}{5} $
✔ $ \frac{3}{5} $

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✔ Answers for Part 2:

a) 6
b) 7
c) 0.8
d) $ \frac{3}{5} $

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Part 3: Calculate the value of the following.

Use the rule: $ \sqrt{a} \times \sqrt{a} = a $

#### a) $ \sqrt{7} \times \sqrt{7} $
- $ \sqrt{7} \times \sqrt{7} = (\sqrt{7})^2 = 7 $
✔ 7

#### b) $ \sqrt{0.3} \times \sqrt{0.3} $
- $ (\sqrt{0.3})^2 = 0.3 $
✔ 0.3

#### c) $ \sqrt{\frac{2}{5}} \times \sqrt{\frac{2}{5}} $
- $ \left(\sqrt{\frac{2}{5}}\right)^2 = \frac{2}{5} $
✔ $ \frac{2}{5} $

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✔ Answers for Part 3:

a) 7
b) 0.3
c) $ \frac{2}{5} $

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✔ Final Answer Summary:

#### Part 1: Perfect Squares?
- a) NO
- b) YES
- c) NO
- d) YES
- e) NO
- f) NO
- g) YES
- h) YES

#### Part 2: Square Roots
- a) 6
- b) 7
- c) 0.8
- d) $ \frac{3}{5} $

#### Part 3: Calculations
- a) 7
- b) 0.3
- c) $ \frac{2}{5} $

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