Square Roots & Cube Roots Diagram-Based Questions

One face = 1,350 ÷ 6 = 225 cm²

Edge = √225 = 15 cm

Volume = 15³ = 3,375 cm³

Trap: 225 is the area of one face.

Answer: 3,375 cm³

Edge of large cube = ∛10,648 = 22 cm

Small cubes along each edge = 22 ÷ 2 = 11

Number = 11³ = 1,331

Trap: 5,324 divides the volume by 2, not by 2³.

Answer: 1,331

7² = 49 and 8² = 64, so √50 is just above 7

√50 ≈ 7.07

That is point B

Trap: C sits halfway, but 50 is much closer to 49 than to 64.

Answer: B

∛40: 3³ = 27 and 4³ = 64 → about 3.4

√40: 6² = 36 and 7² = 49 → about 6.3

Whole numbers between: 4, 5, 6 → 3

Trap: 2 forgets that 6 is below 6.3.

Answer: 3

Squares end only in 0, 1, 4, 5, 6 or 9

8,463 ends in 3, 9,127 in 7, 6,858 in 8 – all impossible

7,744 = 88² ✓

Trap: ending in 8 is ruled out by the table.

Answer: 7,744

Ends in 7 → root ends in 3

Front part 571: 8³ = 512 ≤ 571 < 729 = 9³ → tens digit 8

Cube root = 83

Trap: 87 keeps the 7 instead of using the table.

Answer: 83

First digit from 8² = 64 → 8

Next digit from 164 × 4 = 656 → 4

Remainder 0, so √7,056 = 84

Trap: 164 is the trial divisor, not the root.

Answer: 84

The number rises to 5 and falls back → five 1s in the root

√123454321 = 11111

Check: 11111² = 123454321 ✓

Trap: 12345 copies the rising digits.

Answer: 11111

Squares or cubes = 8 + 2 + 2 = 12

(1 and 64 are both)

Neither = 100 − 12 = 88

Trap: 86 counts 1 and 64 twice.

Answer: 88

Both = sixth powers: 1, 64, 729 → 3

Squares only = 31 − 3 = 28

Trap: 38 is squares or cubes (31 + 10 − 3).

Answer: 28