Answer :
Alright, let's dive into the exercises step by step.
### Exercise 2.1
1. Find the square of the given numbers:
(a) [tex]\( 19 \)[/tex]
[tex]\[ 19^2 = 19 \times 19 = 361 \][/tex]
(b) [tex]\( 20 \)[/tex]
[tex]\[ 20^2 = 20 \times 20 = 400 \][/tex]
(c) [tex]\( 18 \)[/tex]
[tex]\[ 18^2 = 18 \times 18 = 324 \][/tex]
(d) [tex]\( 35 \)[/tex]
[tex]\[ 35^2 = 35 \times 35 = 1225 \][/tex]
(e) [tex]\( 54 \)[/tex]
[tex]\[ 54^2 = 54 \times 54 = 2916 \][/tex]
(f) [tex]\( 63 \)[/tex]
[tex]\[ 63^2 = 63 \times 63 = 3969 \][/tex]
2. Verify the truth of the following statements:
(a) _If any number has 0, 1, 4, 5, 6, and 9 in the ones place, then that number is a square number._
- This statement is False. While many square numbers end in these digits (e.g., 0, 1, 4, 5, 6, 9), not all numbers ending in these digits are square numbers. For example, 14 and 29 end in 4 and 9 respectively but are not squares.
(b) _If there are zeros at the end of any even number, then that number is a square number._
- This statement is False. Having zeros at the end doesn't necessarily make a number a square. For example, 2000 is even and ends in zeros but is not a perfect square.
(c) _0169000 is a perfect square number._
- This statement is False. The number 0169000 simplifies to 169000. The prime factorization of 169000 shows that it is not a perfect square since not all primes are paired.
3. Find the square root of the following number by prime factorization method:
Let's take [tex]\( 169000 \)[/tex] as an example.
Step-by-Step Prime Factorization of 169000:
1. Determine the prime factors:
- Divide by 2:
[tex]\[ 169000 \div 2 = 84500 \quad \text{(2 is a factor)} \][/tex]
[tex]\[ 84500 \div 2 = 42250 \quad \text{(2 is a factor)} \][/tex]
[tex]\[ 42250 \div 2 = 21125 \quad \text{(2 is a factor)} \][/tex]
- Now 21125 is not divisible by 2. Check for the next prime number:
[tex]\[ 21125 \div 5 = 4225 \quad \text{(5 is a factor)} \][/tex]
[tex]\[ 4225 \div 5 = 845 \quad \text{(5 is a factor)} \][/tex]
[tex]\[ 845 \div 5 = 169 \quad \text{(5 is a factor)} \][/tex]
- Now 169 is not divisible by 2 or 5. Check for the next prime number:
[tex]\[ 169 \div 13 = 13 \quad \text{(13 is a factor)} \][/tex]
[tex]\[ 13 \div 13 = 1 \quad \text{(13 is a factor)} \][/tex]
2. Compile the prime factors:
[tex]\[ 169000 = 2^3 \times 5^3 \times 13^2 \][/tex]
3. Group the prime factors in pairs (for square rooting):
[tex]\[ \sqrt{169000} = \sqrt{(2^3 \times 5^3 \times 13^2)} \][/tex]
4. Extract one factor from each pair:
Since [tex]\(2^3\)[/tex] and [tex]\(5^3\)[/tex] do not have pairs of two, we can stop as they will not form a perfect square.
Therefore, 169000 is not a perfect square because it doesn't have all prime factors occurring in pairs.
This concludes our detailed step-by-step solution.
### Exercise 2.1
1. Find the square of the given numbers:
(a) [tex]\( 19 \)[/tex]
[tex]\[ 19^2 = 19 \times 19 = 361 \][/tex]
(b) [tex]\( 20 \)[/tex]
[tex]\[ 20^2 = 20 \times 20 = 400 \][/tex]
(c) [tex]\( 18 \)[/tex]
[tex]\[ 18^2 = 18 \times 18 = 324 \][/tex]
(d) [tex]\( 35 \)[/tex]
[tex]\[ 35^2 = 35 \times 35 = 1225 \][/tex]
(e) [tex]\( 54 \)[/tex]
[tex]\[ 54^2 = 54 \times 54 = 2916 \][/tex]
(f) [tex]\( 63 \)[/tex]
[tex]\[ 63^2 = 63 \times 63 = 3969 \][/tex]
2. Verify the truth of the following statements:
(a) _If any number has 0, 1, 4, 5, 6, and 9 in the ones place, then that number is a square number._
- This statement is False. While many square numbers end in these digits (e.g., 0, 1, 4, 5, 6, 9), not all numbers ending in these digits are square numbers. For example, 14 and 29 end in 4 and 9 respectively but are not squares.
(b) _If there are zeros at the end of any even number, then that number is a square number._
- This statement is False. Having zeros at the end doesn't necessarily make a number a square. For example, 2000 is even and ends in zeros but is not a perfect square.
(c) _0169000 is a perfect square number._
- This statement is False. The number 0169000 simplifies to 169000. The prime factorization of 169000 shows that it is not a perfect square since not all primes are paired.
3. Find the square root of the following number by prime factorization method:
Let's take [tex]\( 169000 \)[/tex] as an example.
Step-by-Step Prime Factorization of 169000:
1. Determine the prime factors:
- Divide by 2:
[tex]\[ 169000 \div 2 = 84500 \quad \text{(2 is a factor)} \][/tex]
[tex]\[ 84500 \div 2 = 42250 \quad \text{(2 is a factor)} \][/tex]
[tex]\[ 42250 \div 2 = 21125 \quad \text{(2 is a factor)} \][/tex]
- Now 21125 is not divisible by 2. Check for the next prime number:
[tex]\[ 21125 \div 5 = 4225 \quad \text{(5 is a factor)} \][/tex]
[tex]\[ 4225 \div 5 = 845 \quad \text{(5 is a factor)} \][/tex]
[tex]\[ 845 \div 5 = 169 \quad \text{(5 is a factor)} \][/tex]
- Now 169 is not divisible by 2 or 5. Check for the next prime number:
[tex]\[ 169 \div 13 = 13 \quad \text{(13 is a factor)} \][/tex]
[tex]\[ 13 \div 13 = 1 \quad \text{(13 is a factor)} \][/tex]
2. Compile the prime factors:
[tex]\[ 169000 = 2^3 \times 5^3 \times 13^2 \][/tex]
3. Group the prime factors in pairs (for square rooting):
[tex]\[ \sqrt{169000} = \sqrt{(2^3 \times 5^3 \times 13^2)} \][/tex]
4. Extract one factor from each pair:
Since [tex]\(2^3\)[/tex] and [tex]\(5^3\)[/tex] do not have pairs of two, we can stop as they will not form a perfect square.
Therefore, 169000 is not a perfect square because it doesn't have all prime factors occurring in pairs.
This concludes our detailed step-by-step solution.
