Answer
**Simplified Answers:**
1. **1.1 Which of the following numbers is rational?**
- **2** and **3** are rational numbers.
2. **12.1 Simplify:**
\[
121 \times \frac{3^{x} + 3^{x+2}}{3^{x+3}} = \frac{1210}{27}
\]
*(Alternatively, \( 44 \frac{22}{27} \).)*
3. **12.2 Simplify:**
\[
\sqrt{0.06 y^{4} + 0.1 y^{4}} = 0.4y^{2}
\]
4. **12.3 Simplify:**
\[
1233(x - 3)(x + 3) - (x - 1)^{2} = 1232x^{2} + 2x - 11,098
\]
Solution
Sure, let's address each part of your questions step by step.
---
### **QUESTION I**
#### **1.1 Which of the following numbers is rational?**
The numbers given are:
- \(2\)
- \(3\)
- \(\sqrt{-25}\)
- \(\pi\)
- \(\sqrt[8]{17}\)
**Rational numbers** are numbers that can be expressed as a fraction \(\frac{a}{b}\) where \(a\) and \(b\) are integers, and \(b \neq 0\).
- **\(2\)**: Rational (can be written as \(\frac{2}{1}\)).
- **\(3\)**: Rational (can be written as \(\frac{3}{1}\)).
- **\(\sqrt{-25}\)**: This is a complex number (\(5i\)), not a rational number.
- **\(\pi\)**: Irrational (it cannot be expressed as a simple fraction).
- **\(\sqrt[8]{17}\)**: Irrational (17 is not a perfect 8th power).
**Answer:** **2** and **3** are rational numbers.
---
### **12. Simplify the following expressions**
#### **12.1 Simplify:**
\[
121 \times \frac{3^{x} + 3^{x+2}}{3^{x+3}}
\]
**Step-by-Step Solution:**
1. **Simplify the numerator:**
\[
3^{x} + 3^{x+2} = 3^{x} + 3^{x} \times 3^{2} = 3^{x}(1 + 9) = 10 \times 3^{x}
\]
2. **Simplify the denominator:**
\[
3^{x+3} = 3^{x} \times 3^{3} = 27 \times 3^{x}
\]
3. **Divide the numerator by the denominator:**
\[
\frac{10 \times 3^{x}}{27 \times 3^{x}} = \frac{10}{27}
\]
4. **Multiply by 121:**
\[
121 \times \frac{10}{27} = \frac{1210}{27}
\]
**Final Simplified Form:**
\[
\frac{1210}{27}
\]
*(This can also be expressed as \( 44 \frac{22}{27} \) if preferred.)*
---
#### **12.2 Simplify:**
\[
\sqrt{0.06 y^{4} + 0.1 y^{4}}
\]
**Step-by-Step Solution:**
1. **Combine like terms:**
\[
0.06 y^{4} + 0.1 y^{4} = 0.16 y^{4}
\]
2. **Take the square root:**
\[
\sqrt{0.16 y^{4}} = \sqrt{0.16} \times \sqrt{y^{4}} = 0.4 \times y^{2} = 0.4y^{2}
\]
**Final Simplified Form:**
\[
0.4y^{2}
\]
---
#### **12.3 Simplify:**
\[
1233(x - 3)(x + 3) - (x - 1)^{2}
\]
**Step-by-Step Solution:**
1. **Expand \((x - 3)(x + 3)\):**
\[
(x - 3)(x + 3) = x^{2} - 9
\]
2. **Expand \((x - 1)^{2}\):**
\[
(x - 1)^{2} = x^{2} - 2x + 1
\]
3. **Substitute back into the original expression:**
\[
1233(x^{2} - 9) - (x^{2} - 2x + 1)
\]
4. **Distribute the constants:**
\[
1233x^{2} - 1233 \times 9 - x^{2} + 2x - 1
\]
5. **Calculate \(1233 \times 9\):**
\[
1233 \times 9 = 11,097
\]
6. **Combine like terms:**
\[
(1233x^{2} - x^{2}) + 2x - (11,097 + 1) = 1232x^{2} + 2x - 11,098
\]
**Final Simplified Form:**
\[
1232x^{2} + 2x - 11,098
\]
---
If you have any more questions or need further clarification on any of these steps, feel free to ask!
Reviewed and approved by the UpStudy tutoring team
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