Cube roots

The cube root of a number is the value whose cube gives it. The symbol is $\sqrt[3]{\,}$. So $\sqrt[3]{125} = 5$ because $5^3 = 125$. While square roots come in $\pm$ pairs, cube roots are always unique because the cube of a positive number is positive and the cube of a negative number is negative.

Concept

If $n = k^3$, then $\sqrt[3]{n} = k$. The most reliable technique for whole numbers is prime factorisation in triples.

Take $5832$. Factor it:

$5832 = 2 \times 2 \times 2 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 = 2^3 \times 3^6.$

We need the primes to come in groups of $3$. They do , $2^3$ is one triple, and $3^6 = (3^2)^3$ is two triples of $3$. Take one factor from each triple:

$\sqrt[3]{5832} = 2 \times 3 \times 3 = 18.$

If after factoring some prime does not fall into a perfect group of three, the number is not a perfect cube. Example: $200 = 2^3 \times 5^2$. The $5$ appears only twice. To turn $200$ into a perfect cube we would need one more $5$: $200 \times 5 = 1000 = 10^3$.

Here is a delightful two-step shortcut that works for any cube of a number from $11$ to $99$.

1. Split the digits of the cube into two parts: the last three digits (units, tens, hundreds) and what is left.
2. The last digit of the cube root comes from the last digit of the cube (using the table from the previous section).
3. The first digit of the cube root comes from the largest cube $\le$ the remaining part.

Example: $\sqrt[3]{19683}$. Split as $19$ and $683$.

• $683$ ends in $3$, so the last digit of the answer is $7$ (since $7^3 = 343$ ends in $3$).
• The largest cube $\le 19$ is $8 = 2^3$, so the first digit is $2$.
• Answer: $27$. Check: $27^3 = 19683$. ✓

This shortcut feels almost like magic but it is just digit arithmetic.

Worked examples

Example 1. Find $\sqrt[3]{1728}$ by prime factorisation.

• $1728 = 2^6 \times 3^3$.
• Group in triples: $(2^3)(2^3)(3^3)$.
• Take one from each: $2 \times 2 \times 3 = 12$.
• So $\sqrt[3]{1728} = 12$.

Example 2. Is $250$ a perfect cube? If not, find the smallest number by which it must be multiplied to become a perfect cube.

• $250 = 2 \times 5^3$. The $2$ appears once.
• We need two more $2$'s to make a triple.
• Multiply by $4$: $250 \times 4 = 1000 = 10^3$.

Example 3. Find $\sqrt[3]{32768}$ using the shortcut.

• Split: $32$ and $768$.
• $768$ ends in $8$. From the table, $2^3 = 8$ ends in $8$. So the last digit of the root is $2$.
• The largest cube $\le 32$ is $27 = 3^3$. So the first digit is $3$.
• Answer: $32$. Check: $32^3 = 32768$. ✓

Example 4. A cubical tank holds $2744$ litres. Find the length of one edge in dm.

• $1$ litre $= 1\text{ dm}^3$, so the edge length is $\sqrt[3]{2744}$.
• $2744 = 2^3 \times 7^3$, so $\sqrt[3]{2744} = 2 \times 7 = 14$.
• Edge length is $14$ dm.

Try it yourself

1. Find $\sqrt[3]{27000}$.
2. Find $\sqrt[3]{9261}$ using the shortcut.
3. Is $1352$ a perfect cube? Justify.
4. By what smallest number should $675$ be multiplied to be a perfect cube?
5. By what smallest number should $1024$ be divided to be a perfect cube?
6. Find $\sqrt[3]{0.027}$ and $\sqrt[3]{-216}$.
7. Find three consecutive whole numbers whose cubes add up to $216$.
8. Estimate $\sqrt[3]{100}$ between two consecutive whole numbers.

Activity

Build cube boxes. Get $64$ small wooden or sugar cubes. Try to arrange them into a perfect cube , you should get a $4 \times 4 \times 4$ stack, demonstrating $4^3 = 64$. Now remove cubes until you are left with $27$ and check that they form a $3 \times 3 \times 3$. Try to do the same with $50$ cubes , you cannot. This physically shows why $50$ is not a perfect cube.