Question 4: Problem solving 4.1 Determine the HCF of $$ x $$ and $$ x^{3} $$ 4.2 Determine the LCM of $$ y $$ and $$ 2 y^{2} $$ [Total: Section A $$ =26 $$ poin SECTION B: INTEGERS Question 5: Operations with integers Calculate the following, without a calculator using prime factors: $$ 5.1 \sqrt[3]{-5832} $$ $$ 5.2 \sqrt{128} $$ Question 6: Operations with integers Calculate the following, without a calculator and show all your calculations: $$ 6.13-4 imes 6+2 \div 2 $$ $$ 6.2 \sqrt[3]{\left(4^{2}-3(2+1) ight)^{2}+3 imes 5} $$

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Question 4: Problem solving 4.1 Determine the HCF of $$ x $$ and $$ x^{3} $$ 4.2 Determine the LCM of $$ y $$ and $$ 2 y^{2} $$ [Total: Section A $$ =26 $$ poin SECTION B: INTEGERS Question 5: Operations with integers Calculate the following, without a calculator using prime factors: $$ 5.1 \sqrt[3]{-5832} $$ $$ 5.2 \sqrt{128} $$ Question 6: Operations with integers Calculate the following, without a calculator and show all your calculations: $$ 6.13-4 	imes 6+2 \div 2 $$ $$ 6.2 \sqrt[3]{\left(4^{2}-3(2+1)ight)^{2}+3 	imes 5} $$

UpStudy AI · Accepted Answer

### Question 4: Problem Solving

#### 4.1 Determine the HCF of $$ x $$ and $$ x^{3} $$

- Since $$ x $$ divides $$ x^{3} $$, the highest common factor (HCF) is the lower power.
- Thus, the HCF is  
  $$
  x
  $$

#### 4.2 Determine the LCM of $$ y $$ and $$ 2y^{2} $$

- For the numerical coefficient, note that one term has coefficient 1 and the other has 2. Thus, the LCM of 1 and 2 is $$2$$.
- For the variable $$ y $$, we take the highest power present, which is $$ y^{2} $$.
- Therefore, the LCM is  
  $$
  2y^{2}
  $$

---

### SECTION B: INTEGERS

#### Question 5: Operations with Integers

##### 5.1 Calculate $$\sqrt[3]{-5832}$$

- Factorize $$5832$$:
  - Divide by 2 three times:  
    $$
    5832 \div 2 = 2916,\quad 2916 \div 2 = 1458,\quad 1458 \div 2 = 729.
    $$
  - Recognize that $$729 = 3^{6}$$.  
- So,  
  $$
  5832 = 2^{3} \times 3^{6}.
  $$
- Taking the cube root (and including the negative sign):  
  $$
  \sqrt[3]{-5832} = -\sqrt[3]{2^{3} \times 3^{6}} = -\left(2 \times 3^{2}\right) = -2 \times 9 = -18.
  $$

##### 5.2 Calculate $$\sqrt{128}$$

- Factorize $$128$$:
  - $$128 = 64 \times 2$$, with $$64 = 8^{2}$$.
- Thus,  
  $$
  \sqrt{128} = \sqrt{64 \times 2} = \sqrt{64} \times \sqrt{2} = 8\sqrt{2}.
  $$

---

#### Question 6: Operations with Integers

##### 6.1 Calculate $$3 - 4 \times 6 + 2 \div 2$$

- Follow the order of operations (multiplication and division before addition and subtraction):
  - Compute multiplication:  
    $$
    4 \times 6 = 24.
    $$
  - Compute division:  
    $$
    2 \div 2 = 1.
    $$
- Substitute back into the expression:  
  $$
  3 - 24 + 1.
  $$
- Perform addition and subtraction:  
  $$
  3 - 24 = -21,\quad -21 + 1 = -20.
  $$

##### 6.2 Calculate $$\sqrt[3]{\left(4^{2} - 3(2+1)\right)^{2} + 3 \times 5}$$

- First, simplify inside the parentheses:
  - Compute $$4^{2}$$:  
    $$
    4^{2} = 16.
    $$
  - Compute $$3(2+1)$$:  
    $$
    3(2+1) = 3 \times 3 = 9.
    $$
  - Subtract:  
    $$
    16 - 9 = 7.
    $$
- Square the result:  
  $$
  7^{2} = 49.
  $$
- Compute $$3 \times 5$$:  
  $$
  3 \times 5 = 15.
  $$
- Add these values inside the cube root:  
  $$
  49 + 15 = 64.
  $$
- Taking the cube root:  
  $$
  \sqrt[3]{64} = 4.
  $$
  
---

### Final Answers

- **4.1 HCF of $$x$$ and $$x^{3}$$:** $$ x $$
- **4.2 LCM of $$y$$ and $$2y^{2}$$:** $$ 2y^{2} $$
- **5.1 $$\sqrt[3]{-5832}$$:** $$-18$$
- **5.2 $$\sqrt{128}$$:** $$8\sqrt{2}$$
- **6.1 $$3 - 4 \times 6 + 2 \div 2$$:** $$-20$$
- **6.2 $$\sqrt[3]{\left(4^{2} - 3(2+1)\right)^{2} + 3 \times 5}$$:** $$4$$
- **4.1 HCF of $$x$$ and $$x^{3}$$:** $$ x $$ - **4.2 LCM of $$y$$ and $$2y^{2}$$:** $$ 2y^{2} $$ - **5.1 $$\sqrt[3]{-5832}$$:** $$-18$$ - **5.2 $$\sqrt{128}$$:** $$8\sqrt{2}$$ - **6.1 $$3 - 4 \times 6 + 2 \div 2$$:** $$-20$$ - **6.2 $$\sqrt[3]{\left(4^{2} - 3(2+1)\right)^{2} + 3 \times 5}$$:** $$4$$

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