Question

1. simplify – find the perfect squares.

1² = ___ 6² = _ 11² = ___
2² = ___ 7² = _ 12² = ___
3² = ___ 8² = _ 13² = ___
4² = ___ 9² = _ 14² = ___
5² = ___ 10² = _ 15² = ___

2.
16² = ___ 21² = ___
17² = ___ 22² = ___
18² = ___ 23² = ___
19² = ___ 24² = ___
20² = ___ 25² = ___

1. plot the following on the number line.

2² −√4 1² √9
number line: −3, −2, −1, 0, 1, 2, 3, 4, 5

1. state the two consecutive whole numbers that the square root lies between:

√77 decimal: ____ consecutive whole #s: ____
√205 decimal: ____ consecutive whole #s: __ & ____

1. rectangle diagram: 3 in by 3 in

find the perimeter of the rectangle:
find the area of the rectangle:

1. the area of a square window is 36 in².

a. what is the side length of the window?
b. what is the perimeter of the window?
window diagram

Answer:

Problem 1: Simplify – Find the perfect squares

Step 1: Calculate \(1^2\)

\(1^2 = 1 \times 1 = 1\)

Step 2: Calculate \(2^2\)

\(2^2 = 2 \times 2 = 4\)

Step 3: Calculate \(3^2\)

\(3^2 = 3 \times 3 = 9\)

Step 4: Calculate \(4^2\)

\(4^2 = 4 \times 4 = 16\)

Step 5: Calculate \(5^2\)

\(5^2 = 5 \times 5 = 25\)

Step 6: Calculate \(6^2\)

\(6^2 = 6 \times 6 = 36\)

Step 7: Calculate \(7^2\)

\(7^2 = 7 \times 7 = 49\)

Step 8: Calculate \(8^2\)

\(8^2 = 8 \times 8 = 64\)

Step 9: Calculate \(9^2\)

\(9^2 = 9 \times 9 = 81\)

Step 10: Calculate \(10^2\)

\(10^2 = 10 \times 10 = 100\)

Step 11: Calculate \(11^2\)

\(11^2 = 11 \times 11 = 121\)

Step 12: Calculate \(12^2\)

\(12^2 = 12 \times 12 = 144\)

Step 13: Calculate \(13^2\)

\(13^2 = 13 \times 13 = 169\)

Step 14: Calculate \(14^2\)

\(14^2 = 14 \times 14 = 196\)

Step 15: Calculate \(15^2\)

\(15^2 = 15 \times 15 = 225\)

Problem 2: Simplify – Find the perfect squares

Step 1: Calculate \(16^2\)

\(16^2 = 16 \times 16 = 256\)

Step 2: Calculate \(17^2\)

\(17^2 = 17 \times 17 = 289\)

Step 3: Calculate \(18^2\)

\(18^2 = 18 \times 18 = 324\)

Step 4: Calculate \(19^2\)

\(19^2 = 19 \times 19 = 361\)

Step 5: Calculate \(20^2\)

\(20^2 = 20 \times 20 = 400\)

Step 6: Calculate \(21^2\)

\(21^2 = 21 \times 21 = 441\)

Step 7: Calculate \(22^2\)

\(22^2 = 22 \times 22 = 484\)

Step 8: Calculate \(23^2\)

\(23^2 = 23 \times 23 = 529\)

Step 9: Calculate \(24^2\)

\(24^2 = 24 \times 24 = 576\)

Step 10: Calculate \(25^2\)

\(25^2 = 25 \times 25 = 625\)

Problem 3: Plot on the number line

First, simplify each expression:

• \(2^2 = 4\)
• \(-\sqrt{4} = -2\) (since \(\sqrt{4} = 2\))
• \(1^2 = 1\)
• \(\sqrt{9} = 3\) (since \(3^2 = 9\))

Now, plot these values on the number line:

• \(-\sqrt{4} = -2\) (plot at \(-2\))
• \(1^2 = 1\) (plot at \(1\))
• \(\sqrt{9} = 3\) (plot at \(3\))
• \(2^2 = 4\) (plot at \(4\))

Problem 4: Consecutive whole numbers for square roots

Part 1: \(\sqrt{77}\)

1. Find the perfect squares around \(77\):

\(8^2 = 64\) and \(9^2 = 81\) (since \(64 < 77 < 81\))

1. Thus, \(\sqrt{77}\) lies between \(8\) and \(9\).
2. Decimal approximation: \(\sqrt{77} \approx 8.77\) (calculated as \(\sqrt{77} \approx 8.77496\))

Part 2: \(\sqrt{205}\)

1. Find the perfect squares around \(205\):

\(14^2 = 196\) and \(15^2 = 225\) (since \(196 < 205 < 225\))

1. Thus, \(\sqrt{205}\) lies between \(14\) and \(15\).
2. Decimal approximation: \(\sqrt{205} \approx 14.32\) (calculated as \(\sqrt{205} \approx 14.3178\))

Problem 5: Perimeter and Area of Rectangle

The rectangle has length \(3\) in and width \(3\) in (it is a square, a special rectangle).

Perimeter of Rectangle:

Formula: \(P = 2 \times (\text{length} + \text{width})\)
Substitute: \(P = 2 \times (3 + 3) = 2 \times 6 = 12\) in

Area of Rectangle:

Formula: \(A = \text{length} \times \text{width}\)
Substitute: \(A = 3 \times 3 = 9\) in²

Problem 6: Square Window (Area = \(36\) in²)

Part (a): Side Length

For a square, \(\text{Area} = \text{side}^2\). Let \(s\) = side length.
\(s^2 = 36\)
Take square root: \(s = \sqrt{36} = 6\) in

Part (b): Perimeter

For a square, \(\text{Perimeter} = 4 \times \text{side}\)
Substitute: \(P = 4 \times 6 = 24\) in

Final Answers

Problem 1:

\(1^2 = \boldsymbol{1}\), \(2^2 = \boldsymbol{4}\), \(3^2 = \boldsymbol{9}\), \(4^2 = \boldsymbol{16}\), \(5^2 = \boldsymbol{25}\),
\(6^2 = \boldsymbol{36}\), \(7^2 = \boldsymbol{49}\), \(8^2 = \boldsymbol{64}\), \(9^2 = \boldsymbol{81}\), \(10^2 = \boldsymbol{100}\),
\(11^2 = \boldsymbol{121}\), \(12^2 = \boldsymbol{144}\), \(13^2 = \boldsymbol{169}\), \(14^2 = \boldsymbol{196}\), \(15^2 = \boldsymbol{225}\)

Problem 2:

\(16^2 = \boldsymbol{256}\), \(17^2 = \boldsymbol{289}\), \(18^2 = \boldsymbol{324}\), \(19^2 = \boldsymbol{361}\), \(20^2 = \boldsymbol{400}\),
\(21^2 = \boldsymbol{441}\), \(22^2 = \boldsymbol{484}\), \(23^2 = \boldsymbol{529}\), \(24^2 = \boldsymbol{576}\), \(25^2 = \boldsymbol{625}\)

Problem 3:

Plotted values: \(-\sqrt{4} = -2\), \(1^2 = 1\), \(\sqrt{9} = 3\), \(2^2 = 4\) (on the number line).

Problem 4:

• \(\sqrt{77}\): Decimal \(\approx \boldsymbol{8.77}\), between \(\boldsymbol{8}\) and \(\boldsymbol{9}\)
• \(\sqrt{205}\): Decimal \(\approx \boldsymbol{14.32}\), between \(\boldsymbol{14}\) and \(\boldsymbol{15}\)

Problem 5:

• Perimeter: \(\boldsymbol{12}\) in
• Area: \(\boldsymbol{9}\) in²

Problem 6:

• (a) Side length: \(\boldsymbol{6}\) in
• (b) Perimeter: \(\boldsymbol{24}\) in