QUESTION I 1.1 Which of the following numbens is rational? $$ 2,3, \sqrt{-25} ; \pi ; \sqrt[8]{17} $$ 12 Simplify: $$ 121 \frac{3^{x}+3^{x+2}}{3^{x+3}} $$ $$ 12.2 \sqrt{0,06 y^{4}+0,1 y^{4}} $$ $$ 1233(x-3)(x+3)-(x-1)^{2} $$

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### **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.

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### **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.)*

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#### **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:**
$$
0.4y^{2}
$$

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#### **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:**
$$
1232x^{2} + 2x - 11,098
$$

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If you have any more questions or need further clarification on any of these steps, feel free to ask!
**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:** $$ 1233(x - 3)(x + 3) - (x - 1)^{2} = 1232x^{2} + 2x - 11,098 $$

QUESTION I
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12 Simplify:

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