Olympiad Test: Cube And Cube Roots - 1

Here 256 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 243 = 3 × 3× 3 × 3 × 3
1331 = 11 × 11 × 11 
250 = 2 × 5 × 5 × 5
Clearly 1331 is a perfect cube.

Volume of cube = (12)³ = 1728 m³

We have 
576 = 2 × 2 × 2 × 3 × 2 × 2 × 2 × 3
∴ 576 should be divided by 9, to get a prefect cube. 

= 784.

1³ + 2³ + 3³ + …… + 9³ = (1 + 2 + 3 …… + 9)²
⇒ (1³ + 3³ + 5³ + 7³ + 9³) + 2³ (1³ + 2³
+ 3³ + 4³)
= (45)² = 2025
⇒ x + 2³ (1 + 2 + 3 + 4)² = 2025
⇒ x = 2025 – 8 × 100 = 1225 

Here 12² + 16² = 400

(3n + 2)³ = 27n³ + 8 + 3 × 3n × 2(3n + 2)
= 27n³ + 8 + 18n (3n + 2)
= 27n³ + 54n² + 36n + 8
= 3 (9n³ + 18n² + 12n + 2) + 2
= 3n + 2

If, 0 < p < 1,
then, p³  < p.

If,       p > 1
then,  p – 1 > 0
∴   p³ > p

Given (2k)³ + (3k)³ + (4k)³ = 2763
⇒ 99 k³ = 2763
⇒ k³ = 27
⇒ k = 3
∴ Numbers are 6, 9, 12.
∴ Their sum = 6 + 9 + 12 = 27

We have 

513 is not a perfect cube.

4.2m = (74.088m³)^1/3

Here 3087 = 3 × 3 × 7 × 7 × 7 = 3² × 7³.
∴ 3087 has two 3 as its factors, but one 3, is short to make 3087, a perfect cube.
∴ 3 should be multiplied to 3087 to produce a perfect cube.

8788 = 2 × 2 × 13 × 13 × 13.
∴ 8788 should be divided by (2 × 2), i.e., 4 to produce quotient as a perfect cube. 

392 = 2 × 2 × 2  × 7 × 7
∴ 392 should be divided by (7 × 7), i.e., 49 to produce quotient, i.e., 8 as a perfect cube. 

2197 = 13³.
∴ The number, i.e., 2197 is odd.
∴ Its cube root should be odd. 

∴ 1278 is even
∴ Its cube root should be even.