Work through the levels in order. Type your answer and press Check — open the worked solution if you get stuck. Answers like -6, 3/4 or 0.25 are typed exactly as shown. A ⭐ Bonus Challenge at the end is extra credit — skip it if the rest was hard work.
Level 1 · Warm-up
1 Solve x3 + 30 = 3.
Show worked solution
x3 = 3 − 30 = −27, so x = ∛(−27) = −3. Check: −27 + 30 = 3 ✓. −9 divides −27 by 3 instead of cube-rooting; 3 drops the sign; ∛33 adds the 30 instead of subtracting it.
2 Solve each equation.
(a) 4x3 = 256
(b) −x3 = 1000
(a) 4x3 = 256
(b) −x3 = 1000
Show worked solution
(a) x3 = 64, so x = 4. (b) x3 = −1000, so x = −10. Check: 4 × 64 = 256 ✓ and −(−10)3 = −(−1000) = 1000 ✓.
3 A cube-shaped box has a volume of 8000 cm3. What is the area of one face, in cm2?
Show worked solution
203 = 8000, so the side is 20 cm and one face is 20 × 20 = 400 cm2.
Level 2 · Core
4 Solve (2x − 1)3 = 729.
Show worked solution
2x − 1 = ∛729 = 9, so 2x = 10 and x = 5. Check: (10 − 1)3 = 93 = 729 ✓. 4 subtracts the 1 instead of adding it; 9 stops at 2x − 1; 365 ignores the cube and solves 2x − 1 = 729.
5 (a) Solve (x − 2)2 = 49.
(b) Solve (x − 2)3 = −64.
(b) Solve (x − 2)3 = −64.
Show worked solution
(a) x − 2 = 7 or x − 2 = −7, so the smaller root is −5 and the larger root is 9. (b) x − 2 = ∛(−64) = −4, so x = −2, the only solution. Check: (−4)3 = −64 ✓.
6 64 identical small cubes, each with edges of 3 cm, are stacked to make one large solid cube. How long is an edge of the large cube, in cm?
Show worked solution
64 = 43, so each edge of the large cube is 4 small cubes long: 4 × 3 = 12 cm. Check: total volume 64 × 27 = 1728 = 123 ✓.
Level 3 · Challenge
7 Three statements:
1 2x3 = 16 and (2x)3 = 16 have the same solution.
2 If k is positive, x3 = −k has exactly one real solution.
3 If (x − 3)3 = −1, then x = 4.
Which statements are correct?
1 2x3 = 16 and (2x)3 = 16 have the same solution.
2 If k is positive, x3 = −k has exactly one real solution.
3 If (x − 3)3 = −1, then x = 4.
Which statements are correct?
Show worked solution
2x3 = 16 gives x3 = 8 and x = 2, but (2x)3 = 16 gives 8x3 = 16, x3 = 2 and x = ∛2 (1 ✗). Every number has exactly one real cube root, so x = −∛k is the only solution (2 ✓). x − 3 = −1, so x = 2, not 4 (3 ✗). 2 only.
8 Solve −2(3 − x)3 = 250. Give the value of 3 − x, then x.
Show worked solution
(3 − x)3 = 250 ÷ (−2) = −125, so 3 − x = −5 and x = 3 + 5 = 8. Check: −2 × (3 − 8)3 = −2 × (−125) = 250 ✓.
9 When each edge of a cube is made 5 cm longer, its volume becomes 8 times as large. How long was each edge at first, in cm?
Show worked solution
(x + 5)3 = 8x3 = (2x)3, so x + 5 = 2x and x = 5. Check: 53 = 125 and 103 = 1000 = 8 × 125 ✓.