Find the cube root of 5 1182 2197 .

The given number is 5 1182 2197 . Now, 5 1182 2197 3 = 12167 2197 3 = 12167 3 2197 3 By Prime factorisation, we have 23 12167 23 529 23 23 1 13 2197 13 169 13 13 1 Expressing 12167 and 2197 as prime factors, we have 12167 = 23 × 23 × 23 2197 = 13 × 13 × 13 Thus, 5 1182 2197 3 = 12167 3 2197 3 = 23 × 23 × 23 3 13 × 13 × 13 3 = 23 13 = 1 10 13 Hence, the cube root of 5 1182 2197 is 1 10 13 .

Find the cube root of 5 1182 2197 .

The given number is 5 1182 2197 . Now, 5 1182 2197 3 = 12167 2197 3 = 12167 3 2197 3 By Prime factorisation, we have 23 12167 23 529 23 23 1 13 2197 13 169 13 13 1 Expressing 12167 and 2197 as prime factors, we have 12167 = 23 × 23 × 23 2197 = 13 × 13 × 13 Thus, 5 1182 2197 3 = 12167 3 2197 3 = 23 × 23 × 23 3 13 × 13 × 13 3 = 23 13 = 1 10 13 Hence, the cube root of 5 1182 2197 is 1 10 13 .

Evaluate: 512 × 729 3

Given expression: 512 × 729 3 Now, 512 × 729 3 = 512 3 × 729 3 By Prime factorisation, we have 3 729 3 243 3 81 3 27 3 9 3 3 1 2 512 2 256 2 128 2 64 2 32 2 16 2 8 2 4 2 2 1 Expressing 729 and 512 as prime factors, we have 729 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 512 = 3 × 3 × 3 × 3 × 3 × 3 Thus, 512 × 729 3 = 512 3 × 729 3 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 3 × 3 × 3 × 3 × 3 × 3 × 3 3 = ( 2 × 2 × 2 ) × ( 3 × 3 ) = 72 Hence, the required answer is 72 .

Evaluate: ( - 1331 ) × ( 3375 ) 3

Given expression: ( - 1331 ) × ( 3375 ) 3 = - 1331 3 × 3375 3 = - 1331 3 × 3375 3 By Prime factorisation, we have 3 3375 3 1125 3 375 5 125 5 25 5 5 1 11 1331 11 121 11 11 1 Expressing 1331 and 3375 as prime factors, we have 1331 = 11 × 11 × 11 3375 = 3 × 3 × 3 × 5 × 5 × 5 Thus, ( - 1331 ) × ( 3375 ) 3 = - 1331 3 × 3375 3 = - 11 × 11 × 11 3 × 3 × 3 × 3 × 5 × 5 × 5 3 = - 11 × 3 × 5 = - 165 Hence, the required answer is - 165 .

Evaluate: 27 3 + 0 . 008 3 + 0 . 064 3 (Express simplified answer in decimal)

Given expression: 27 3 + 0 . 008 3 + 0 . 064 3 = 27 3 + 8 1000 3 + 64 1000 3 = 27 3 + 8 3 1000 3 + 64 3 1000 3 = 3 × 3 × 3 3 + 2 × 2 × 2 3 10 × 10 × 10 3 + 4 × 4 × 4 3 10 × 10 × 10 3 = 3 + 2 10 + 4 10 = 3 + 0 . 2 + 0 . 4 = 3 . 6 Hence, the required answer is 3 . 6 .