A square root asks for a side. A cube root asks for an edge.
Find the value
A cubical water tank holds 216 cubic feet. How long is each edge?
A cube's volume is edge × edge × edge, so you need the number that gives 216 when multiplied by itself three times: 6 × 6 × 6 = 216, so the edge is 6 ft. C divides by 3 and D divides by 2 — but volume is not shared out between the edges, it is built from them by multiplying. A is the answer to a different question: 4 × 4 × 4 is only 64.
Just as a perfect square is built from two.
The small ³ is read "cubed", and it comes from exactly this picture — three dimensions, three factors.
| n | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
|---|---|---|---|---|---|---|---|---|---|---|
| n³ | 1 | 8 | 27 | 64 | 125 | 216 | 343 | 512 | 729 | 1000 |
Learn this row. Every cube root question at this level is one of these ten numbers read backwards. Just past it: 11³ = 1,331 and 12³ = 1,728.
3√216 = 6, because 6³ = 216.
The little 3 tucked into the root sign is what makes it a cube root. With no number there, as in √36, it means a square root.
3√64 is 4, not 8. The number 64 happens to be both a perfect square (8²) and a perfect cube (4³), and reaching for the familiar square root is easy. Read the root sign before you answer.
Useful when the number is not one you recognise.
3√1728
not on the table
1728 = 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3
split into primes
= (2 × 2 × 3) × (2 × 2 × 3) × (2 × 2 × 3)
three identical groups
3√1728 = 12
one group
Check: 12 × 12 = 144, and 144 × 12 = 1728 ✓ — always worth the ten seconds.
3√−125 = −5, because (−5)(−5)(−5) = −125.
Squaring can never produce a negative, so √−25 has no answer among the real numbers. Cubing can, so 3√−125 is a perfectly ordinary −5. An odd number of negative factors keeps the sign negative.
A cube's edge is the cube root of its volume. Every other measurement starts from the edge.
A cubical water tank has a volume of 729 cubic decimetres. What is the area of one face, and of all six faces?
edge = 3√729 = 9 dm
9 × 9 × 9 = 729
one face = 9 × 9 = 81 dm²
a face is a square
six faces = 6 × 81 = 486 dm²
a cube has six equal faces
729 ÷ 6 = 121.5 mixes cubic units with square units and means nothing. The volume only tells you the edge; the faces have to be built from the edge, 9 × 9 at a time.
Scaling: a cube has an edge of 5 cm. A second cube has 27 times its volume. How long is the second cube's edge?
125 × 27 = 3,375 cm³
first volume 5³ = 125
3√3,375 = 15 cm
15 × 15 × 15 = 3,375
Quicker: 3√27 = 3, so the edge is 3 times as long, 3 × 5 = 15 cm. Volume grows by the cube of the edge's growth, never by the same factor.
The same "three equal factors" question; these are the forms that look harder than they are.
Find 3√8125.
3√8 = 2 and 3√125 = 5
root the top and the bottom separately
3√8125 = 25
check: 25 × 25 × 25 = 8125 ✓
Both parts must be perfect cubes for the answer to come out as a neat fraction.
Find 3√0.064.
0.4 × 0.4 × 0.4 = 0.064
4³ = 64, with three decimal places
3√0.064 = 0.4
one decimal place, not three
Each factor of 0.4 brings one decimal place, so three places in the cube means one place in the root.
Solve x³ = −216.
x = 3√−216
undo the cube
x = −6
(−6)(−6)(−6) = −216
Unlike x² = 36, which has two answers, a cube equation has exactly one real answer: 6³ is +216, so only −6 works.
Trap the number between two cubes you know.
3√100
100 is not in the table
4³ = 64 and 5³ = 125
the two cubes either side
4 < 3√100 < 5
and nearer 5, since 100 is nearer 125
Cube roots are usually tested as a root to evaluate (often of a negative or a fraction), a "between which two whole numbers", or a cube's volume turned into its edge and then its faces. The fastest route is the cube table up to 10³; the 10-second check is to multiply your answer by itself three times.
Simplify
3√−512 + √81
8³ = 512, so 3√−512 = −8, and √81 = 9. Then −8 + 9 = 1. D gets the two the wrong way round. B and C add 8 and 9 instead of subtracting — they lose the minus sign that a cube root is allowed to keep.
Hint: do the two roots separately before combining them, and watch which one is allowed to be negative.
