QUESTION 2 2.1 Consider the following: $$ 3^{3} ;-\sqrt[3]{27} ; 50 ; 3 \pi ; \sqrt{\frac{4}{49}} ; \sqrt{3} ; \sqrt{-4} ;-2,14 ;(\sqrt{3})^{2} $$ Choose from the list: 2.1.1 An integer with factors 2 and 5. (1) 2.1.2 An integer but not a whole number. (1) 2.1.3 A natural number smaller than 10 . (1) 2.1.5 All the irrational numbers. (2) 2.2 Between which 2 whole numbers does $$ \sqrt[3]{1113} $$ lie? 2.3 Determine the LCM of $$ 150 ; 250 $$ and 375. (4)

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**2.1 Analysis of the List**

Given list:  
$$
3^{3};\quad -\sqrt[3]{27};\quad 50;\quad 3\pi;\quad \sqrt{\frac{4}{49}};\quad \sqrt{3};\quad \sqrt{-4};\quad -2,14; \quad (\sqrt{3})^{2}
$$

First, simplify each element:
- $$3^3 = 27$$.  
- $$-\sqrt[3]{27} = -3$$ (since $$\sqrt[3]{27} = 3$$).  
- $$50$$ stays $$50$$.  
- $$3\pi$$ remains $$3\pi$$.  
- $$\sqrt{\frac{4}{49}} = \frac{2}{7}$$ (since $$\sqrt{4} = 2$$ and $$\sqrt{49} = 7$$).  
- $$\sqrt{3}$$ is irrational.  
- $$\sqrt{-4}$$ is not a real number (it is imaginary).  
- $$-2,14$$ is interpreted as $$-2.14$$.  
- $$(\sqrt{3})^2 = 3$$.

**2.1.1 An integer with factors 2 and 5**

An integer with factors 2 and 5 has at least a factor of 2 and a factor of 5 in its prime factorization.  
- $$50 = 2 \times 5^2$$.

Thus, the answer is:  
$$
50.
$$

---

**2.1.2 An integer but not a whole number**

A whole number is a nonnegative integer. Thus, an integer that is not whole must be negative.  
- $$-\sqrt[3]{27} = -3$$ is an integer and negative.

Thus, the answer is:  
$$
-3.
$$

---

**2.1.3 A natural number smaller than 10**

Natural numbers are positive integers. The candidates are those that are positive integers less than 10.  
- $$(\sqrt{3})^2 = 3$$ is a natural number smaller than 10.

Thus, the answer is:  
$$
3.
$$

---

**2.1.5 All the irrational numbers**

From the list, identify the irrational numbers:  
- $$3\pi$$ is irrational (since $$\pi$$ is irrational).  
- $$\sqrt{3}$$ is irrational.

Other numbers are either rational or not real.

Thus, the set of irrational numbers is:  
$$
\{3\pi,\, \sqrt{3}\}.
$$

---

**2.2 Between which 2 whole numbers does $$ \sqrt[3]{1113} $$ lie?**

Find the cube numbers around 1113:  
- $$10^3 = 1000$$.  
- $$11^3 = 1331$$.

Since $$1000 < 1113 < 1331$$, it follows that  
$$
\sqrt[3]{1113} \text{ lies between } 10 \text{ and } 11.
$$

Thus, the answer is:  
$$
10 \text{ and } 11.
$$

---

**2.3 Determine the LCM of $$150$$, $$250$$ and $$375$$**

First, factor each number into primes:
- $$150 = 2 \times 3 \times 5^2$$.  
- $$250 = 2 \times 5^3$$.  
- $$375 = 3 \times 5^3$$.

The LCM is obtained by taking the highest power of each prime present:
- For $$2$$: highest power is $$2^1$$.  
- For $$3$$: highest power is $$3^1$$.  
- For $$5$$: highest power is $$5^3$$.

Thus, the LCM is:
$$
2^1 \times 3^1 \times 5^3 = 2 \times 3 \times 125 = 750.
$$

Thus, the answer is:  
$$
750.
$$
**2.1 Analysis of the List** Given list: $$ 3^{3},\quad -\sqrt[3]{27},\quad 50,\quad 3\pi,\quad \sqrt{\frac{4}{49}},\quad \sqrt{3},\quad \sqrt{-4},\quad -2.14,\quad (\sqrt{3})^{2} $$ Simplified list: $$ 27,\quad -3,\quad 50,\quad 3\pi,\quad \frac{2}{7},\quad \sqrt{3},\quad \text{Not Real},\quad -2.14,\quad 3 $$ **2.1.1 An integer with factors 2 and 5** - **Answer:** 50 **2.1.2 An integer but not a whole number** - **Answer:** -3 **2.1.3 A natural number smaller than 10** - **Answer:** 3 **2.1.5 All the irrational numbers** - **Answer:** $$3\pi$$ and $$\sqrt{3}$$ **2.2 Between which 2 whole numbers does $$ \sqrt[3]{1113} $$ lie?** - **Answer:** 10 and 11 **2.3 Determine the LCM of 150, 250, and 375** - **Answer:** 750

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