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**QUESTION 1**
1. **Rational Number:** A number that can be expressed as a fraction of two integers, like \( \frac{1}{2} \) or \( 4.75 \).
2. **Values of \( x \):**
- **Non-real:** \( x = 16 \) makes the expression inside the square root negative.
- **Irrational Number:** \( x = -4 \) results in an irrational number.
- **Rational Number:** \( x = 4 \) yields a rational number.
3. **Perfect Square:** Any integer that is the square of another integer, such as 16, 25, or 36.
Solución
### **QUESTION 1**
#### **1.1 Define the term: Rational Number**
A **rational number** is any number that can be expressed as the quotient or fraction \( \frac{p}{q} \) of two integers, where:
- \( p \) (the numerator) is an integer.
- \( q \) (the denominator) is a non-zero integer.
In other words, a rational number can be written as a simple or terminating decimal, or as a repeating decimal. Examples of rational numbers include \( \frac{1}{2} \), \( -3 \), and \( 4.75 \) (which can be written as \( \frac{19}{4} \)).
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#### **1.2 For which values of \( x, \ x \in \{-4, 4, 16\} \), is \( \sqrt{\frac{25}{13 - x}} \)**
##### **1.2.1 Non-real?**
A square root is **non-real** (i.e., it does not produce a real number) when the expression inside the square root is negative.
Consider the expression inside the square root:
\[
\frac{25}{13 - x}
\]
For the expression to be negative:
\[
\frac{25}{13 - x} < 0
\]
Since \( 25 \) is positive, the denominator must be negative:
\[
13 - x < 0 \implies x > 13
\]
**Given that \( x \in \{-4, 4, 16\} \):**
- \( x = 16 \) satisfies \( 16 > 13 \)
**Therefore, \( \sqrt{\frac{25}{13 - x}} \) is non-real when \( x = 16 \).**
##### **1.2.2 An Irrational Number?**
The expression \( \sqrt{\frac{25}{13 - x}} \) is irrational if the result cannot be expressed as a ratio of two integers.
Simplify the expression:
\[
\sqrt{\frac{25}{13 - x}} = \frac{5}{\sqrt{13 - x}}
\]
For the expression to be rational, \( \sqrt{13 - x} \) must be rational, which occurs if \( 13 - x \) is a perfect square.
**Evaluate for each \( x \):**
- **\( x = -4 \):**
\[
13 - (-4) = 17 \quad (\sqrt{17} \text{ is irrational})
\]
- **\( x = 4 \):**
\[
13 - 4 = 9 \quad (\sqrt{9} = 3 \text{ is rational})
\]
- **\( x = 16 \):**
\[
13 - 16 = -3 \quad (\text{Non-real, as covered in 1.2.1})
\]
**Therefore, \( \sqrt{\frac{25}{13 - x}} \) is irrational when \( x = -4 \).**
##### **1.2.3 A Rational Number?**
From the analysis in **1.2.2**, the expression is rational when \( \sqrt{13 - x} \) is rational, which happens if \( 13 - x \) is a perfect square.
**From the previous evaluations:**
- **\( x = 4 \):**
\[
13 - 4 = 9 \quad (\sqrt{9} = 3 \text{ is rational})
\]
**Therefore, \( \sqrt{\frac{25}{13 - x}} \) is a rational number when \( x = 4 \).**
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#### **1.3 Write Down Any Perfect Square Greater Than 10 but Less Than 40**
A **perfect square** is an integer that is the square of another integer.
**Identifying perfect squares between 10 and 40:**
- \( 4^2 = 16 \)
- \( 5^2 = 25 \)
- \( 6^2 = 36 \)
**Examples:**
- **16** is a perfect square (\( 4^2 \)) greater than 10 but less than 40.
- Alternatively, **25** (\( 5^2 \)) and **36** (\( 6^2 \)) also satisfy the condition.
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