Squares and Square Roots | rosario_d2023 | Live - Free Printable

Problem Analysis and Solution

The worksheet focuses on squares, square roots, and their properties. Let's solve each section step by step.

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#### Section 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. For example, \(16 = 4^2\) is a perfect square, but \(15\) is not.

- a) 120
- Check if there exists an integer \(n\) such that \(n^2 = 120\).
- The closest perfect squares are \(11^2 = 121\) and \(10^2 = 100\). Since \(120\) is not a perfect square, the answer is NO.

- b) 144
- Check if there exists an integer \(n\) such that \(n^2 = 144\).
- \(12^2 = 144\), so \(144\) is a perfect square. The answer is YES.

- c) 160
- Check if there exists an integer \(n\) such that \(n^2 = 160\).
- The closest perfect squares are \(12^2 = 144\) and \(13^2 = 169\). Since \(160\) is not a perfect square, the answer is NO.

- d) 196
- Check if there exists an integer \(n\) such that \(n^2 = 196\).
- \(14^2 = 196\), so \(196\) is a perfect square. The answer is YES.

- e) 250
- Check if there exists an integer \(n\) such that \(n^2 = 250\).
- The closest perfect squares are \(15^2 = 225\) and \(16^2 = 256\). Since \(250\) is not a perfect square, the answer is NO.

- f) 300
- Check if there exists an integer \(n\) such that \(n^2 = 300\).
- The closest perfect squares are \(17^2 = 289\) and \(18^2 = 324\). Since \(300\) is not a perfect square, the answer is NO.

- g) 324
- Check if there exists an integer \(n\) such that \(n^2 = 324\).
- \(18^2 = 324\), so \(324\) is a perfect square. The answer is YES.

- h) 484
- Check if there exists an integer \(n\) such that \(n^2 = 484\).
- \(22^2 = 484\), so \(484\) is a perfect square. The answer is YES.

Final Answers for Section 1:
- a) NO
- b) YES
- c) NO
- d) YES
- e) NO
- f) NO
- g) YES
- h) YES

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#### Section 2: Determine the square roots of the following.

The square root of a number \(x\) is a value \(y\) such that \(y^2 = x\).

- a) \(\sqrt{36}\)
- The square root of \(36\) is \(6\) because \(6^2 = 36\).
- The correct answer is 6.

- b) \(\sqrt{7^2}\)
- The square root of \(7^2\) is \(7\) because \((7^2)^{1/2} = 7\).
- The correct answer is 7.

- c) \(\sqrt{0.64}\)
- The square root of \(0.64\) is \(0.8\) because \(0.8^2 = 0.64\).
- The correct answer is 0.8.

- d) \(\sqrt{\frac{9}{25}}\)
- The square root of \(\frac{9}{25}\) is \(\frac{\sqrt{9}}{\sqrt{25}} = \frac{3}{5}\).
- The correct answer is \(\frac{3}{5}\).

Final Answers for Section 2:
- a) 6
- b) 7
- c) 0.8
- d) \(\frac{3}{5}\)

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#### Section 3: Calculate the value of the following.

Use the property that \(\sqrt{a} \times \sqrt{b} = \sqrt{a \times b}\).

- a) \(\sqrt{7} \times \sqrt{7}\)
- Using the property, \(\sqrt{7} \times \sqrt{7} = \sqrt{7 \times 7} = \sqrt{49} = 7\).
- The correct answer is 7.

- b) \(\sqrt{0.3} \times \sqrt{0.3}\)
- Using the property, \(\sqrt{0.3} \times \sqrt{0.3} = \sqrt{0.3 \times 0.3} = \sqrt{0.09} = 0.3\).
- The correct answer is 0.3.

- c) \(\sqrt{\frac{2}{5}} \times \sqrt{\frac{2}{5}}\)
- Using the property, \(\sqrt{\frac{2}{5}} \times \sqrt{\frac{2}{5}} = \sqrt{\frac{2}{5} \times \frac{2}{5}} = \sqrt{\frac{4}{25}} = \frac{2}{5}\).
- The correct answer is \(\frac{2}{5}\).

Final Answers for Section 3:
- a) 7
- b) 0.3
- c) \(\frac{2}{5}\)

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Final Answer Summary:

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

2. Square Roots:
- a) 6
- b) 7
- c) 0.8
- d) \(\frac{3}{5}\)

3. Calculations:
- a) 7
- b) 0.3
- c) \(\frac{2}{5}\)

Boxed Final Answer:
\[
\boxed{
\begin{array}{l}
\text{Section 1: } \text{NO, YES, NO, YES, NO, NO, YES, YES} \\
\text{Section 2: } 6, 7, 0.8, \frac{3}{5} \\
\text{Section 3: } 7, 0.3, \frac{2}{5}
\end{array}
}
\]