Show that each of the following numbers is a perfect square. Also find

1.	Show that each of the following numbers is a perfect square. Also find the number whose square is the given number in each case:(i) 1156(ii) 2025(iii) 14641
Answer» (i) 1156 The prime factors for 1156 1156 = 2×2×17×17 = (2×2) × (17×17) (none of the prime factors are left out..) Hence, 1156 is a perfect square. Now, 1156 = (2×17) × (2×17) = 34 × 34 = (34)² ∴ 1156 is a square of 34 (ii) 2025 The prime factors for 2025 2025 = 3×3×3×3×5×5 = (3×3) × (3×3) × (5×5) (none of the prime factors are left out..) Hence, 2025 is a perfect square. Now, 2025 = (3×3×5) × (3×3×5) = 45 × 45 = (45)² ∴ 2025 is a square of 45. (iii) 14641 The prime factors for 14641 14641 = 11×11×11×11 = (11×11) × (11×11) (none of the prime factors are left out..) Hence, 14641 is a perfect square. Now, 14641 = (11×11) × (11×11) = 121 × 121 = (121)² ∴ 14641 is a square of 121.

Show that each of the following numbers is a perfect square. Also find the number whose square is the given number in each case:(i) 1156(ii) 2025(iii) 14641

Answer»

(i) 1156

The prime factors for 1156

1156 = 2×2×17×17

= (2×2) × (17×17) (none of the prime factors are left out..)

Hence, 1156 is a perfect square.

Now,

1156 = (2×17) × (2×17)

= 34 × 34

= (34)²

∴ 1156 is a square of 34

(ii) 2025

The prime factors for 2025

2025 = 3×3×3×3×5×5

= (3×3) × (3×3) × (5×5) (none of the prime factors are left out..)

Hence, 2025 is a perfect square.

Now,

2025 = (3×3×5) × (3×3×5)

= 45 × 45

= (45)²

∴ 2025 is a square of 45.

(iii) 14641

The prime factors for 14641

14641 = 11×11×11×11

= (11×11) × (11×11) (none of the prime factors are left out..)

Hence, 14641 is a perfect square.

Now,

14641 = (11×11) × (11×11)

= 121 × 121

= (121)²

∴ 14641 is a square of 121.