Chapter 6.5 — Exercise 6.4 — Cubes and Cube Roots

Cube of Even and Odd Numbers

The parity rule for squares carries over unchanged: the cube of an even number is even, and the cube of an odd number is odd. Unlike squaring, though, cubing keeps track of sign too — cubing a negative number gives a negative result (since three negative factors multiply to a negative), which never happens with squares.

The Units-Digit Pattern for Cubes

Cubes have their own units-digit table, different from the squares' version — notably, 2, 3, 7 and 8 (impossible endings for squares) are all perfectly valid endings for cubes:

Notice 2 and 8 swap, and 3 and 7 swap — cubing a number ending in 2 gives a cube ending in 8, and vice versa. This pairing becomes the basis for quickly finding a cube root's units digit by estimation later in this chapter.

Computing Cubes Directly

8³ = 8×8×8 = 512. 16³ = 16×16×16 = 4096. 21³ = 21×21×21 = 9261. 30³ = 30×30×30 = 27,000 — worth double-checking on numbers ending in zero, since it's easy to drop a zero when multiplying three times in a row.

Testing Whether a Number Is a Perfect Cube

The test mirrors the square version exactly, but groups prime factors in threes instead of twos:

• 243 = 3×3×3×3×3 — five 3's, which don't group evenly into threes (one 3 left over) → not a perfect cube.
• 516 = 2×2×3×43 — no prime appears three times at all → not a perfect cube.
• 729 = 3×3×3×3×3×3 — six 3's, grouping perfectly into two groups of three → is a perfect cube (9³ = 729).
• 8000 = 2×2×2×2×2×2×5×5×5 — six 2's and three 5's, both grouping evenly → is a perfect cube (20³ = 8000).
• 2700 = 2×2×3×3×3×5×5 — the 2's and 5's don't reach a full group of three → not a perfect cube.

Finding the Smallest Adjustment to Make a Perfect Cube

Once the prime factorisation reveals which primes are short of a full group of three, fixing it is direct:

• 8788 = 2×2×13×13×13 — the factor 2 appears only twice, one short of a group of three. Multiplying by 2 completes the group.
• 7803 = 3×3×3×17×17 — the factor 17 appears only twice. Multiplying by 17 completes it.
• 8640 = 2×2×2×2×2×2×3×3×3×5 — the 2's and 3's already group into threes perfectly, but the single 5 doesn't. Dividing by 5 removes the unpaired factor entirely, leaving a perfect cube.

Two Word Problems

A cuboid of dimensions 12 cm × 8 cm × 3 cm, built from unit cubes, needs 12×8×3 = 288 unit cubes — no cube-root reasoning needed here at all, just the volume formula, since the cuboid isn't itself a cube.

A cleverer question: what's the smallest prime dividing 3¹¹ + 5¹³? Rather than computing either enormous power, notice that any odd number multiplied by itself repeatedly stays odd, so both 3¹¹ and 5¹³ are odd. The sum of two odd numbers is always even, so 3¹¹ + 5¹³ is even — meaning 2 divides it, without needing to know the actual value at all.

Why Cubes Allow Endings That Squares Forbid

It's worth understanding why 2, 3, 7 and 8 are impossible endings for squares but perfectly fine for cubes. A square's units digit only depends on the units digit of the original number squared, and checking each digit 0 through 9 squared shows none happens to land on 2, 3, 7 or 8 — it's simply a fact about which remainders arise from squaring, not a deeper rule. Cubing involves an extra multiplication, which shuffles the possible units digits differently — every digit 0 through 9, cubed, produces a different and complete spread across all ten possible endings, which is exactly why the cube units-digit table pairs every digit with exactly one other (or itself), with no digit left permanently excluded the way squares exclude four of them.

Cubes Grow Much Faster Than Squares

It's worth pausing on just how quickly cubes outpace squares as the base number grows. 10² = 100, but 10³ = 1000 — ten times larger, not just a bit more. By the time the base reaches 100, the square is 10,000 while the cube is 1,000,000 — a hundred times larger. This gap widens indefinitely as the base grows, which is exactly why cube-related quantities (volumes) become impractically large so much faster than square-related ones (areas) as an object's size scales up — a fact with real consequences well beyond this exercise, in engineering and biology alike, wherever volume and surface area are compared as something scales up or down.

Twos and Threes — Where They Get Confused

• Grouping prime factors in twos instead of threes. The perfect-cube test needs groups of three matching factors — applying the perfect-square grouping here gives a wrong conclusion.
• Dropping a digit when cubing a large number or one ending in zero. Multiplying three times in a row compounds any small slip — double-check results like 30³ = 27,000 rather than trusting mental arithmetic alone.
• Assuming every leftover factor needs multiplying. Sometimes dividing out the unpaired factor (as with 8640's stray 5) is the right fix, not multiplying by more of it.

The Odd-Number-Sum Trick Has No Simple Cube Equivalent

Squares had a neat shortcut — the sum of the first n odd numbers equals n² — and it's natural to wonder whether cubes have something similarly clean. They do, though it looks different: the sum of the first n cubes (1³+2³+3³+...+n³) always equals (1+2+3+...+n)², the square of the sum of the first n whole numbers. For n=3: 1³+2³+3³ = 1+8+27 = 36, and (1+2+3)² = 6² = 36 — matching exactly. This is a genuinely striking fact in its own right (the sum of cubes always turns out to be a perfect square), even though it isn't the focus of this particular exercise — a reminder that the number patterns explored around squares and cubes rarely stop at the one fact being tested in a given question — noticing the extra pattern is often more memorable than the specific answer the question originally asked for, and it's exactly this kind of curiosity that turns a single exercise into a broader understanding of how these numbers behave.

Structure That Leads to Actual Roots

Recognising perfect cubes and their prime-factor structure here leads directly into Exercise 6.5, which finds actual cube roots by both prime factorisation and a faster estimation method built on the units-digit pairing introduced above. For the parallel square-root reasoning this exercise builds on, revisit Exercise 6.2.