4th Summative Test in Mathematics 7, Quarter 1, Week 7 & 8, from the Department of Education, Region III, Schools Division of Zambales.

Problem Analysis and Solution

The provided image is a summative test in Mathematics for Grade 7, focusing on concepts related to numbers, roots, and approximations. Below is a detailed explanation and solution for each question.

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#### Question 1:
What do you call a number that can be expressed in the form \(\frac{a}{b}\), where \(a\) and \(b\) are integers, and \(b\) is not equal to 0?

- Options:
- a. Integers
- b. Irrational
- c. Natural
- d. Rational

Solution:
A number that can be expressed as \(\frac{a}{b}\), where \(a\) and \(b\) are integers and \(b \neq 0\), is called a rational number. This is the definition of a rational number.

Answer: d. Rational

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#### Question 2:
Which of the following refers to a number whose decimal representation is neither terminating nor repeating? This number cannot be expressed as a quotient of integers.

- Options:
- a. Integers
- b. Irrational
- c. Natural
- d. Rational

Solution:
A number whose decimal representation is neither terminating nor repeating and cannot be expressed as a quotient of integers is called an irrational number. Examples include \(\sqrt{2}\), \(\pi\), and \(e\).

Answer: b. Irrational

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#### Question 3:
What do you call the positive nth root of a number?

- Options:
- a. Perfect square
- b. Principal root
- c. Radical
- d. Radicand

Solution:
The positive \(n\)th root of a number is called the principal root. For example, the principal square root of 9 is 3, not -3.

Answer: b. Principal root

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#### Question 4:
What is the principal root of \(\sqrt{25}\)?

- Options:
- a. 2
- b. 5
- c. 25
- d. 50

Solution:
The principal root of \(\sqrt{25}\) is the positive square root of 25, which is 5.

Answer: b. 5

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#### Question 5:
What is the principal root of \(\sqrt{10}\)?

- Options:
- a. 3.162277660...
- b. 5
- c. 10
- d. 100

Solution:
The principal root of \(\sqrt{10}\) is the positive square root of 10, which is approximately 3.162277660... (an irrational number).

Answer: a. 3.162277660...

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#### Question 6:
Which of the following describes the principal root of \(\sqrt{50}\)?

- Options:
- a. Integers
- b. Irrational
- c. Natural
- d. Rational

Solution:
The principal root of \(\sqrt{50}\) is the positive square root of 50, which is approximately 7.071067811... (an irrational number).

Answer: b. Irrational

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#### Question 7:
Which of the numbers is classified as a perfect square integer?

- Options:
- a. 6
- b. 9
- c. 12
- d. 20

Solution:
A perfect square integer is a number that can be expressed as the square of an integer. Among the options:
- \(6 = \sqrt{6}\) (not a perfect square)
- \(9 = 3^2\) (perfect square)
- \(12 = \sqrt{12}\) (not a perfect square)
- \(20 = \sqrt{20}\) (not a perfect square)

Answer: b. 9

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#### Question 8:
Between what two consecutive integers does the square root of 18 lie?

- Options:
- a. 3 & 4
- b. 4 & 5
- c. 5 & 6
- d. 16 & 25

Solution:
To find the range of \(\sqrt{18}\):
- \(4^2 = 16\)
- \(5^2 = 25\)
Thus, \(\sqrt{18}\) lies between 4 and 5.

Answer: b. 4 & 5

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#### Question 9:
Find the number whose square root lies between 5 and 6.

- Options:
- a. 24
- b. 25
- c. 26
- d. 36

Solution:
The square root of a number lies between 5 and 6 if the number is between \(5^2 = 25\) and \(6^2 = 36\). Among the options:
- 24 < 25
- 25 = \(5^2\) (not between 5 and 6)
- 26 is between 25 and 36
- 36 = \(6^2\) (not between 5 and 6)

Answer: c. 26

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#### Question 10:
Find the square root of 16.

- Options:
- a. \(\pm 4\)
- b. \(\pm 5\)
- c. \(\pm 6\)
- d. \(\pm 8\)

Solution:
The square root of 16 is both 4 and -4, as \(4^2 = 16\) and \((-4)^2 = 16\).

Answer: a. \(\pm 4\)

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#### Question 11:
Which of the following is the perfect square number closest but greater than 136?

- Options:
- a. 100
- b. 138
- c. 144
- d. 169

Solution:
Among the options:
- 100 = \(10^2\) (less than 136)
- 138 is not a perfect square
- 144 = \(12^2\) (greater than 136)
- 169 = \(13^2\) (greater than 136 but not the closest)

The closest perfect square greater than 136 is 144.

Answer: c. 144

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#### Question 12:
Between which consecutive numbers is the estimate \(\sqrt{10}\) lie?

- Options:
- a. 2 and 3
- b. 3 and 4
- c. 4 and 5
- d. 5 and 6

Solution:
To find the range of \(\sqrt{10}\):
- \(3^2 = 9\)
- \(4^2 = 16\)
Thus, \(\sqrt{10}\) lies between 3 and 4.

Answer: b. 3 and 4

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#### Question 13:
Which of the following is the closest approximation of \(\sqrt{51}\)?

- Options:
- a. 6.64
- b. 6.94
- c. 7.04
- d. 7.14

Solution:
To approximate \(\sqrt{51}\):
- \(7^2 = 49\)
- \(8^2 = 64\)
Thus, \(\sqrt{51}\) is slightly more than 7. Among the options, 7.04 is the closest.

Answer: c. 7.04

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Final Answers:

1. d. Rational
2. b. Irrational
3. b. Principal root
4. b. 5
5. a. 3.162277660...
6. b. Irrational
7. b. 9
8. b. 4 & 5
9. c. 26
10. a. \(\pm 4\)
11. c. 144
12. b. 3 and 4
13. c. 7.04

Boxed Final Answer:
\[
\boxed{
\begin{array}{ll}
1. & \text{d. Rational} \\
2. & \text{b. Irrational} \\
3. & \text{b. Principal root} \\
4. & \text{b. 5} \\
5. & \text{a. 3.162277660...} \\
6. & \text{b. Irrational} \\
7. & \text{b. 9} \\
8. & \text{b. 4 & 5} \\
9. & \text{c. 26} \\
10. & \text{a. } \pm 4 \\
11. & \text{c. 144} \\
12. & \text{b. 3 and 4} \\
13. & \text{c. 7.04} \\
\end{array}
}
\]