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Example 14: Evaluate: −2300 × 5290
3
3
Solution: −2300 × 5290 = − 2300 × 5290 =− 23 × 529 × 1000 =− 23 × 529 × 1000 23 529
3
3
3
3
23 23
= − 23 × 23 × 23 × 10 × 10 × 10 = –23 × 10 = –230 1
3
3
Example 15: Find the cube root of 1.331.
.
1 331 × 1000 1331
Solution: 1.331 = =
1000 1000
1331 3 1331 3 11 11 11 11
×
×
.
.
∴ 3 1 331 = 3 1000 = 3 1000 = 3 10 10 10 = 10 = 11
×
×
Example 16: Three numbers are in the ratio 2 : 3 : 4. The sum of their cubes is 33957. Find the numbers.
Solution: Let the numbers be 2x, 3x and 4x, then
3
3
3
3
3
3
(2x) + (3x) + (4x) = 33957 ⇒ 8x + 27x + 64x = 33957
33957
3
⇒ 99x = 33957 ⇒ x = = 343
3
99
3
⇒ x = 7 × 7 × 7 ⇒ x = 7 × 7 × = 7
7
3
Thus, the numbers are 2x = 2 × 7 = 14; 3x = 3 × 7 = 21; and 4x = 4 × 7 = 28
Example 17: Is 3528 a perfect cube? If not, find the smallest natural number by which 3528 must be
multiplied so that the product is a perfect cube. Also, find the cube root of the number obtained.
Solution: Prime factorisation 3528 = 2 × 2 × 2 × 3 × 3 × 7 × 7 = 2 × 3 × 7 2 2 3528
2
3
The prime factors 3 and 7 do not appear in groups of three. 2 1764
Therefore, 3528 is not a perfect cube. 2 3 882
441
To make it a perfect cube, it must be multiplied by 3 × 7. 3 147
So, the smallest number by which 3528 must be multiplied to make the product a perfect 7 49
cube = 3 × 7 = 21. 7 7
1
3528 × (3 × 7) = 2 × 2 × 2 × 3 × 3 × 3 × 7 × 7 × 7, which is a perfect cube.
Hence, the perfect cube is 3528 × 21 = 74088
Therefore, 74088 = 2 × 2 × 2 × 3 × 3 × 3 × 7 × 7 × 7
333××
××
××
3 74088 = 3 222 ×× 777
Thus, 3 74088 = 2 × 3 × 7 = 42
Example 18: Find the smallest number by which 26244 should be divided so that the quotient is
a perfect cube. Also, find the cube root of the product so obtained.
Solution: Prime factorisation of 26244 = 2 × 2 × 3 × 3 × 3 × 3 × 3 × 3 × 3 × 3 2 26244
Grouping them into groups of three, we get 2 13122
2
3
3
26244 = 2 × 2 × (3 × 3 × 3) × (3 × 3 × 3) × 3 × 3 = 2 × 3 × 3 × 3 2 3 6561
3
2187
We can see that the numbers (2 × 2) and (3 × 3) are left ungrouped. 3 729
So, if we divide 26,244 by (2 × 2 × 3 × 3) = 36, it becomes a perfect cube. 3 243
3 81
26244
Also, the quotient = = 729 is a perfect cube 3 27
Therefore, 36 729 = 3 × 3 × 3 × 3 × 3 × 3 3 9
3
3
3 729 = (333× × ) (333 = 3 × 3 = 9 1
×
×
×
)
3
159 Cubes and Cube Roots