AFCAT Numerical Ability · Squares, Cubes and Roots

Cubes, Cube Roots, and Square and Cube Roots Together

Cube a number and use the cube identities, recognise a perfect cube, find cube roots of whole numbers, negatives, decimals and fractions, and evaluate square and cube roots in one expression.

All Squares, Cubes and Roots notes

Why this matters

This is the biggest page of the chapter, and most of it is quick: a cube, a cube root, or a sum of the two kinds of root. Marks go on the sign, since the cube root of a negative number is negative and not undefined, and on decimals, where a cube root takes decimal places in threes.

Concept 1 of 6: Cubing a number, and the cube identities

A cube multiplies a number by itself three times. Unlike a square, it keeps the sign: a negative number cubed is negative.

Definition

$n^3 = n \times n \times n$ : $6^3 = 216$ . Also $0^3 = 0$ and $1^3 = 1$ .
$(-n)^3 = -n^3$ : $(-4)^3 = -64$ .
$(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$ and $(a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3$ .
$a^3 + b^3 = (a + b)(a^2 - ab + b^2)$ and $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$ .

Sum and difference of cubes

a^3 \pm b^3 = (a \pm b)(a^2 \mp ab + b^2)

Worked example

Find

5^3 - 4^3

with the identity, and check it directly.

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionHard

Example 1 · Squares, Cubes and Roots · Cubes and Cube Roots

Using identity:

a^3 - b^3 = (a-b)(a^2 + ab + b^2)

, simplify

(3)^3 - (2)^3

An odd power keeps the sign

(-5)^3 = -125

, but

(-5)^2 = 25

. Cubes of negative numbers are negative.

A cube of a sum is not a sum of cubes

(1 + 4)^3 = 125

, not

1 + 64 = 65

. The expansion has two middle terms.

Drill 6 more on cubing a number, and the cube identities

Concept 2 of 6: The cubes from 1 to 10, and spotting a perfect cube

Learn the cubes from 1 to 10. Unlike squares, cubes end in every digit, and each last digit of a cube comes from exactly one last digit of its root. That makes cube roots of perfect cubes quick.

Definition

A number is a perfect cube when every prime exponent is a multiple of 3. For a perfect cube between 1,000 and 1,000,000 the root has two digits:

Strike off the last three digits. The largest cube at or below what is left gives the tens digit.
The last digit of the number gives the units digit, with no second choice (the table below).
For 2,744: 2 lies between $1^3$ and $2^3$ , so the root is in the teens, and a last digit 4 gives 4. The root is 14.

n	n³	Last digit of n³
1	1	1
2	8	8 2 and 8 swap: a cube ending in 8 has a root ending in 2.
3	27	7 3 and 7 swap: a cube ending in 7 has a root ending in 3.
4	64	4
5	125	5
6	216	6
7	343	3
8	512	2
9	729	9
10	1,000	0

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionModerate

Example 2 · Squares, Cubes and Roots · Cubes and Cube Roots

1728

a perfect cube?

Cubes can end in any digit

A cube can end in 2, 3, 7 or 8:

8^3 = 512

and

7^3 = 343

. The last digit never rules a number out as a cube, unlike a square.

Bracket with cubes, not squares

For 9,261, strike off three digits, not two, and compare 9 with the cubes 8 and 27. Comparing 92 with squares sends you to the wrong tens digit.

Drill 5 more on the cubes from 1 to 10, and spotting a perfect cube

Concept 3 of 6: Cube roots by prime factors

A cube root undoes a cube. Group the prime factors in threes; one factor from each group makes the root.

Definition

$\sqrt[3]{n^3} = n$ and $(\sqrt[3]{a})^3 = a$ .
By factors: divide every exponent by 3. $5832 = 2^3 \times 3^6$ , so $\sqrt[3]{5832} = 2 \times 3^2 = 18$ .
Roots inside roots are worked from the inside out: $\sqrt{\sqrt[3]{729}} = \sqrt 9 = 3$ .

Cube root by prime factors

\sqrt[3]{p^{3a} q^{3b}} = p^{a} q^{b}

Worked example

Find

\sqrt[3]{10648}

by prime factors.

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionHard

Example 3 · Squares, Cubes and Roots · Cubes and Cube Roots

Simplify:

\sqrt[3]{\sqrt[3]{512}}

Groups of three, not pairs

\sqrt[3]{2^6}

2^2 = 4

, not

2^3 = 8

. Divide each exponent by 3 for a cube root, by 2 only for a square root.

Drill 6 more on cube roots by prime factors

Concept 4 of 6: Cube roots of negative numbers, and cube root equations

A cube root keeps the sign, because a negative number cubed is negative. So every number, negative or not, has one real cube root, which a square root cannot offer.

Definition

$\sqrt[3]{-a} = -\sqrt[3]{a}$ : $\sqrt[3]{-125} = -5$ .
A negative number has no real square root, but it always has a real cube root.
$\sqrt[3]{ax + b} = k$ : cube both sides, $ax + b = k^3$ , and solve. $k$ may be negative here.

Clearing a cube root

\sqrt[3]{ax + b} = k \;\Rightarrow\; ax + b = k^3

Worked example

Solve

\sqrt[3]{x + 10} = -2

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionModerate

Example 4 · Squares, Cubes and Roots · Cubes and Cube Roots

Solve:

\sqrt[3]{x - 1} = 3

Cube, do not square

\sqrt[3]{x} = 4

gives

x = 64

, not 16. Clearing a cube root needs the cube of both sides.

A negative cube root exists

\sqrt[3]{-216} = -6

. Answering 'undefined' mixes up cube roots with square roots.

Drill 5 more on cube roots of negative numbers, and cube root equations

Concept 5 of 6: Cube roots of decimals and fractions

For cube roots, decimal places come in threes: the root has a third as many as the number. A fraction's cube root is the cube root of the top over that of the bottom.

Definition

$\sqrt[3]{0.729} = \sqrt[3]{\tfrac{729}{1000}} = \tfrac{9}{10} = 0.9$ .
Every three decimal places in the number make one in the root: $\sqrt[3]{0.000125} = 0.05$ .
$\sqrt[3]{\tfrac{64}{125}} = \tfrac 45$ .

Cube root of a decimal

\sqrt[3]{\dfrac{a}{10^{3k}}} = \dfrac{\sqrt[3]{a}}{10^{k}}

Worked example

Find

\sqrt[3]{0.000343}

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionHard

Example 5 · Squares, Cubes and Roots · Cubes and Cube Roots

What is

\sqrt[3]{0.008}

Decimal places in threes

\sqrt[3]{0.0027}

is not 0.03, because

0.03^3 = 0.000027

. Four decimal places do not give a neat cube root:

\sqrt[3]{0.0027}

is about 0.139.

A third of the places

\sqrt[3]{0.000064} = 0.04

: six places become two. Halving them, as for a square root, gives 0.004.

Drill 3 more on cube roots of decimals and fractions

Concept 6 of 6: Square roots and cube roots in one expression

Evaluate each root on its own, square roots by pairs and cube roots by threes, then do the arithmetic. The small number on the root sign says which root it is.

Definition

A root sign with no small number is a square root; a small 3 makes it a cube root.
Cube roots split over a product just as square roots do: $\sqrt[3]{25} \times \sqrt[3]{5} = \sqrt[3]{125} = 5$ .
$(\sqrt a)^2 = a$ and $(\sqrt[3]{a})^3 = a$ .
A negative number under a cube root gives a negative value; under a square root it gives no real value.

One root at a time

\sqrt{a^2} = a, \quad \sqrt[3]{b^3} = b

Worked example

Find

\sqrt{225} \div \sqrt[3]{125} + \sqrt[3]{-64}

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionModerate

Example 6 · Squares, Cubes and Roots · Cubes and Cube Roots

Simplify:

\sqrt[3]{-27} + \sqrt{16}

Read the small index

\sqrt[3]{64} = 4

but

\sqrt{64} = 8

. Missing the 3 on the root sign doubles the answer here.

Drill 8 more on square roots and cube roots in one expression

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (5)

Cubing a number, and the cube identities
Sum and difference of cubes
$a^3 \pm b^3 = (a \pm b)(a^2 \mp ab + b^2)$
Cube roots by prime factors
Cube root by prime factors
$\sqrt[3]{p^{3a} q^{3b}} = p^{a} q^{b}$
Cube roots of negative numbers, and cube root equations
Clearing a cube root
$\sqrt[3]{ax + b} = k \;\Rightarrow\; ax + b = k^3$
Cube roots of decimals and fractions
Cube root of a decimal
$\sqrt[3]{\dfrac{a}{10^{3k}}} = \dfrac{\sqrt[3]{a}}{10^{k}}$
Square roots and cube roots in one expression
One root at a time
$\sqrt{a^2} = a, \quad \sqrt[3]{b^3} = b$

Reference tables (1)

The cubes from 1 to 10, and spotting a perfect cube10 rows

n	n³	Last digit of n³
1	1	1
2	8	8 2 and 8 swap: a cube ending in 8 has a root ending in 2.
3	27	7 3 and 7 swap: a cube ending in 7 has a root ending in 3.
4	64	4
5	125	5
6	216	6
7	343	3
8	512	2
9	729	9
10	1,000	0

Watch out for (10)

An odd power keeps the sign
→ Cubing a number, and the cube identities
A cube of a sum is not a sum of cubes
→ Cubing a number, and the cube identities
Cubes can end in any digit
→ The cubes from 1 to 10, and spotting a perfect cube
Bracket with cubes, not squares
→ The cubes from 1 to 10, and spotting a perfect cube
Groups of three, not pairs
→ Cube roots by prime factors
Cube, do not square
→ Cube roots of negative numbers, and cube root equations
A negative cube root exists
→ Cube roots of negative numbers, and cube root equations
Decimal places in threes
→ Cube roots of decimals and fractions
A third of the places
→ Cube roots of decimals and fractions
Read the small index
→ Square roots and cube roots in one expression

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