03 February 2025 1. Given $$ \sqrt{2} ;-3 ; \sqrt{-2} ; \frac{4}{0} ; \frac{0}{6} ; 7 ; \sqrt{9} ; \sqrt[3]{9} ; \sqrt[3]{-27} $$ From the list of the numbors above, give a) All the non-real numbers D) All the integers c) All the rational d) Al the irrational numbers e) All the natural numbers. numbers 2. Simplify the following expressions: a) $$ 2 a b^{2}(4 a-3 c+7 b c) $$ b) $$ (2 a-3 b)\left(4 a^{2}+6 a b+9 b^{2}\right) $$ c) $$ (8 x y-7 z)(9 x y+3 x-5 z) $$ d) $$ (f-15 g)(6 f-4 f g-3 g) $$

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### **Question 1**

Given the list of numbers:
$$ \sqrt{2},\ -3,\ \sqrt{-2},\ \frac{4}{0},\ \frac{0}{6},\ 7,\ \sqrt{9},\ \sqrt[3]{9},\ \sqrt[3]{-27} $$

Let's classify them based on different categories:

1. **a) All the Non-Real Numbers:**
   - **$$\sqrt{-2}$$**: This is an imaginary number since it involves the square root of a negative number.

2. **b) All the Integers:**
   - **$$-3$$**: An integer.
   - **$$0$$**: From $$\frac{0}{6}$$, where $$\frac{0}{6} = 0$$.
   - **$$7$$**: An integer.
   - **$$\sqrt{9} = 3$$**: An integer.
   - **$$\sqrt[3]{-27} = -3$$**: An integer.

**Integers in the list:** $$-3,\ 0,\ 7,\ 3$$

3. **c) All the Rational Numbers:**
   - **$$-3$$**: Can be expressed as $$\frac{-3}{1}$$.
   - **$$0$$**: From $$\frac{0}{6} = 0$$.
   - **$$7$$**: Can be expressed as $$\frac{7}{1}$$.
   - **$$\sqrt{9} = 3$$**: Can be expressed as $$\frac{3}{1}$$.
   - **$$\sqrt[3]{-27} = -3$$**: Can be expressed as $$\frac{-3}{1}$$.

**Rational numbers in the list:** $$-3,\ 0,\ 7,\ 3$$

4. **d) All the Irrational Numbers:**
   - **$$\sqrt{2}$$**: An irrational number.
   - **$$\sqrt[3]{9}$$**: Since $$9$$ is not a perfect cube, its cube root is irrational.

**Irrational numbers in the list:** $$\sqrt{2},\ \sqrt[3]{9}$$

5. **e) All the Natural Numbers:**
   - **$$7$$**
   - **$$3$$**: From $$\sqrt{9} = 3$$

**Natural numbers in the list:** $$7,\ 3$$

---

### **Question 2**

Simplify the following algebraic expressions:

#### **a) $$ 2ab^{2}(4a - 3c + 7bc) $$**
Distribute $$2ab^{2}$$ across each term inside the parentheses:
$$
\begin{align*}
2ab^{2} \cdot 4a &= 8a^{2}b^{2} \
2ab^{2} \cdot (-3c) &= -6ab^{2}c \
2ab^{2} \cdot 7bc &= 14ab^{3}c \
\end{align*}
$$
**Simplified Expression:**
$$
8a^{2}b^{2} - 6ab^{2}c + 14ab^{3}c
$$

#### **b) $$ (2a - 3b)(4a^{2} + 6ab + 9b^{2}) $$**
Recognize that the expression resembles the formula for the difference of cubes:
$$
(a - b)(a^{2} + ab + b^{2}) = a^{3} - b^{3}
$$
Here, $$2a$$ and $$3b$$ are analogous to $$a$$ and $$b$$ respectively.

$$
(2a - 3b)(4a^{2} + 6ab + 9b^{2}) = (2a)^{3} - (3b)^{3} = 8a^{3} - 27b^{3}
$$
**Simplified Expression:**
$$
8a^{3} - 27b^{3}
$$

#### **c) $$ (8xy - 7z)(9xy + 3x - 5z) $$**
Multiply each term in the first parenthesis by each term in the second parenthesis:

$$
\begin{align*}
8xy \cdot 9xy &= 72x^{2}y^{2} \
8xy \cdot 3x &= 24x^{2}y \
8xy \cdot (-5z) &= -40xy z \
-7z \cdot 9xy &= -63xy z \
-7z \cdot 3x &= -21xz \
-7z \cdot (-5z) &= 35z^{2} \
\end{align*}
$$

**Combine Like Terms:**
$$
72x^{2}y^{2} + 24x^{2}y - 103xy z - 21xz + 35z^{2}
$$

**Simplified Expression:**
$$
72x^{2}y^{2} + 24x^{2}y - 103xyz - 21xz + 35z^{2}
$$

#### **d) $$ (f - 15g)(6f - 4fg - 3g) $$**
Multiply each term in the first parenthesis by each term in the second parenthesis:

$$
\begin{align*}
f \cdot 6f &= 6f^{2} \
f \cdot (-4fg) &= -4f^{2}g \
f \cdot (-3g) &= -3fg \
-15g \cdot 6f &= -90fg \
-15g \cdot (-4fg) &= 60fg^{2} \
-15g \cdot (-3g) &= 45g^{2} \
\end{align*}
$$

**Combine Like Terms:**
$$
6f^{2} - 4f^{2}g - 93fg + 60fg^{2} + 45g^{2}
$$

**Simplified Expression:**
$$
6f^{2} - 4f^{2}g - 93fg + 60fg^{2} + 45g^{2}
$$

---

Feel free to ask if you need further clarification on any of these solutions!
### **Question 1** Given the list of numbers: $$ \sqrt{2},\ -3,\ \sqrt{-2},\ \frac{4}{0},\ \frac{0}{6},\ 7,\ \sqrt{9},\ \sqrt[3]{9},\ \sqrt[3]{-27} $$ **a) Non-Real Numbers:** - $$\sqrt{-2}$$ **b) Integers:** - $$-3$$ - $$0$$ (from $$\frac{0}{6}$$) - $$7$$ - $$3$$ ($$\sqrt{9}$$) - $$-3$$ ($$\sqrt[3]{-27}$$) **c) Rational Numbers:** - $$-3$$ - $$0$$ - $$7$$ - $$3$$ - $$-3$$ **d) Irrational Numbers:** - $$\sqrt{2}$$ - $$\sqrt[3]{9}$$ **e) Natural Numbers:** - $$7$$ - $$3$$ ($$\sqrt{9}$$) --- ### **Question 2** **a) Simplify $$ 2ab^{2}(4a - 3c + 7bc) $$:** $$ 8a^{2}b^{2} - 6ab^{2}c + 14ab^{3}c $$ **b) Simplify $$ (2a - 3b)(4a^{2} + 6ab + 9b^{2}) $$:** $$ 8a^{3} - 27b^{3} $$ **c) Simplify $$ (8xy - 7z)(9xy + 3x - 5z) $$:** $$ 72x^{2}y^{2} + 24x^{2}y - 103xyz - 21xz + 35z^{2} $$ **d) Simplify $$ (f - 15g)(6f - 4fg - 3g) $$:** $$ 6f^{2} - 4f^{2}g - 93fg + 60fg^{2} + 45g^{2} $$

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