Exercise Time 3.1
1. Find which of the following numbers are perfect squares by using the prime factorization method:
(a) 196
(b) 1764
(c) 6292
(d) 3675
2. What will be the one place digit of the squares of the following numbers:
1
(a) 334
(b) 372
(c) 12795
(d) 93489
3. Just observe the one place digits and state which of the following are perfect squares:
Ro
(a) 4624
(b) 8343
(c) 6387
(d) 1832
4. Is 256 a perfect square? If so, find the number whose square is 256.
5. Show that 5929 is a perfect square.
6. Find the sum without actual addition :
(a) 1+3+5+7
(b) 1+3+5+ 7 + 9 + 11
7. Fill in the blanks :
(a) 463? - 462
(b) 49? - 48
8. Express 49 as the sum of odd numbers.
9. Express 25 as the sum of odd numbers.
?
Answers
Step-by-step explanation:
Solution:
Resolving 484 as the product of primes, we get
484 = 2 × 2 × 11 × 11
√484 = √(2 × 2 × 11 × 11)
= 2 × 11
Therefore, √484 = 22
2. Find the square root of 324.
Solution:
The square root of 324 by prime factorization, we get
324 = 2 × 2 × 3 × 3 × 3 × 3
√324 = √(2 × 2 × 3 × 3 × 3 × 3)
= 2 × 3 × 3
Therefore, √324 = 18
3. Find out the square root of 1764.
Solution:
The square root of 1764 by prime factorization, we get
1764 = 2 x 2 x 3 x 3 x 7 x 7.
√1764 = √(2 x 2 x 3 x 3 x 7 x 7)
= 2 x 3 x 7
Therefore, √1764 = 42.
4. Evaluate √4356
Solution:
By using prime factorization, we get
4356 = 2 x 2 x 3 x 3 x 11 x 11
√4356 = √(2 x 2 x 3 x 3 x 11 x 11)
= 2 × 3 × 11
Therefore, √4356 = 66.
5. Evaluate √11025
Solution:
By using prime factorization, we get
11025 = 5 x 5 x 3 x 3 x 7 x 7.
√11025 = √(5 x 5 x 3 x 3 x 7 x 7)
= 5 × 3 × 7
Therefore, √11025 = 105
6. In an auditorium, the number of rows is equal to the number of chairs in each row. If the capacity of the auditorium is 2025, find the number of chairs in each row.
Solution:
Let the number of chairs in each row be x.
Then, the number of rows = x.
Total number of chairs in the auditorium = (x × x) = x²
But, the capacity of the auditorium = 2025 (given).
Therefore, x² = 2025
= 5 × 5 × 3 × 3 × 3 × 3
x = (5 × 3 × 3) = 45.
Hence, the number of chairs in each row = 45
7. Find the smallest number by which 396 must be multiplied so that the product becomes a perfect square.
Solution:
By prime factorization, we get
396 = 2 × 2 × 3 × 3 × 11
It is clear that in order to get a perfect square, one more 11 is required.
So, the given number should be multiplied by 11 to make the product a perfect square.
Step-by-step explanation:
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