Find the cube root of - 64 1331 .

The given number is - 64 1331 . Now, - 64 1331 3 = - 64 3 1331 3 = - 64 3 1331 3 By Prime factorisation, we have 2 64 2 32 2 16 2 8 2 4 2 2 1 11 1331 11 121 11 11 1 Expressing 64 and 1331 as prime factors, we have 64 = 2 × 2 × 2 × 2 × 2 × 2 1331 = 11 × 11 × 11 Thus, - 64 1331 3 = - 64 3 1331 3 = - 2 × 2 × 2 × 2 × 2 × 2 3 11 × 11 × 11 3 = - 2 × 2 11 = - 4 11 Hence, the cube root of - 64 1331 is - 4 11 .

Find the cube root of 4 17 27 .

The given number is 4 17 27 . Now, 4 17 27 3 = 125 27 3 = 125 3 27 3 By Prime factorisation, we have 5 125 5 25 5 5 1 3 27 3 9 3 3 1 Expressing 125 and 27 as prime factors, we have 125 = 5 × 5 × 5 27 = 3 × 3 × 3 Thus, 4 17 27 3 = 125 3 27 3 = 5 × 5 × 5 3 3 × 3 × 3 3 = 5 3 = 1 2 3 Hence, the cube root of 4 17 27 is 1 2 3 .

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 .