Cube roots and cube-root expressions

A big Rubik's-style cube is built from $27$ identical small cubes. How many small cubes are there along each edge? A gift box in the shape of a cube has a volume of $2$ litres, that is $2000$ cm³. How long is each edge, in centimetres?

For the first question, $3\cdot3\cdot3=27$ , so each edge has $3$ small cubes. The second question needs a number whose cube is $2000$ . That is a cube root.

A cube with edge

a

and volume

a^3

The cube root of a real number

Explore: the cubes of a few numbers

$x$	$-3$	$-2$	$-1$	$0$	$1$	$2$	$3$
$x^3$	$-27$	$-8$	$-1$	$0$	$1$	$8$	$27$

Unlike a square, the cube of a negative number is negative. Each value in the bottom row appears exactly once. For example, only one number cubes to $-8$ , namely $-2$ .

Remember: the cube root of a positive number is positive, of a negative number is negative, and of $0$ is $0$ . Also, $\sqrt[3]{-a}=-\sqrt[3]{a}$ .

For a number like $2000$ , no whole number cubes to $2000$ (since $12^3=1728$ and $13^3=2197$ ). We use a calculator.

The identities $\left(\sqrt[3]{a}\right)^3=a$ and $\sqrt[3]{a^3}=a$ make many expressions quick to evaluate.

Comparing cube roots

Cubing keeps the order: a larger number has a larger cube. So for all real numbers $a$ and $b$ : $a<b\iff\sqrt[3]{a}<\sqrt[3]{b}.$

Cube-root expressions

Compare with square roots: $\sqrt{A^2}=|A|$ , but $\sqrt[3]{A^3}=A$ , with no absolute value and no condition $A\ge0$ .

Watch out Common mistakes

Thinking a negative number has no cube root. Wrong: " $\sqrt[3]{-8}$ is not defined". Why it is wrong: $(-2)^3=-8$ . Right: $\sqrt[3]{-8}=-2$ .
Writing $\sqrt[3]{a^3}=|a|$ . Wrong: $\sqrt[3]{(-5)^3}=5$ . Why it is wrong: only square roots need the absolute value. Right: $\sqrt[3]{(-5)^3}=-5$ .
Confusing the cube root with dividing by three. Wrong: $\sqrt[3]{27}=9$ . Why it is wrong: $9^3=729$ . Right: $\sqrt[3]{27}=3$ because $3^3=27$ .
Adding a condition that is not needed. Wrong: " $\sqrt[3]{x-2}$ is defined when $x\ge2$ ". Why it is wrong: a cube root is defined for every number. Right: it is defined for every $x$ .

Key terms

English	Tiếng Việt
cube root	căn bậc ba
cube-root expression	căn thức bậc ba
cube of a number	lập phương của một số
cube	hình lập phương
index of the root	chỉ số của căn
volume	thể tích
expression under the root sign	biểu thức lấy căn

Summary

$x=\sqrt[3]{a}\iff x^3=a$ . Every real number has exactly one cube root.
$\left(\sqrt[3]{a}\right)^3=a$ , $\sqrt[3]{a^3}=a$ , $\sqrt[3]{-a}=-\sqrt[3]{a}$ .
$a<b\iff\sqrt[3]{a}<\sqrt[3]{b}$ ; to compare, write both as cube roots or compare their cubes.
$\sqrt[3]{A}$ is defined whenever $A$ is; $\sqrt[3]{A^3}=A$ (no absolute value).
A cube of volume $V$ has edge $\sqrt[3]{V}$ .

Practice

Each exercise comes with what you need and some steps to start. Try it yourself first, then open the solution.

Exercise 1

Evaluate: a) $\sqrt[3]{343}$ ; b) $\sqrt[3]{-0.027}$ ; c) $\sqrt[3]{\dfrac{64}{125}}$ .

What you need and how to start

What you need: $\sqrt[3]{a}=x\iff x^3=a$ .

Steps:

Remember the cubes $1^3$ to $10^3$ .
For a negative number, the cube root is negative.

Show solution

a) Since $7^3=343$ , $\sqrt[3]{343}=7$ . b) Since $(-0.3)^3=-0.027$ , $\sqrt[3]{-0.027}=-0.3$ . c) Since $\left(\dfrac45\right)^3=\dfrac{64}{125}$ , $\sqrt[3]{\dfrac{64}{125}}=\dfrac45$ .

Answer. a) $7$ ; b) $-0.3$ ; c) $\dfrac45$ .

Exercise 2

Evaluate $\sqrt[3]{-216}+\sqrt[3]{1000}-\left(\sqrt[3]{7}\right)^3$ .

What you need and how to start

What you need: $\left(\sqrt[3]{a}\right)^3=a$ .

Steps:

Work out each root.
Add and subtract.

Show solution

We have $\sqrt[3]{-216}+\sqrt[3]{1000}-\left(\sqrt[3]{7}\right)^3=-6+10-7=-3$ .

Answer. $-3$ .

Exercise 3

Use a calculator to find (rounded to two decimal places): a) $\sqrt[3]{50}$ ; b) $\sqrt[3]{-12}$ .

What you need and how to start

What you need: the calculator has a cube-root key; the cube root of a negative number is negative.

Steps:

Use the calculator.
Round to the second decimal place.

Show solution

Using a calculator, we get: a) $\sqrt[3]{50}\approx3.68$ ; b) $\sqrt[3]{-12}\approx-2.29$ .

Answer. a) $3.68$ ; b) $-2.29$ .

Exercise 4

Compare: a) $5$ and $\sqrt[3]{124}$ ; b) $-2$ and $\sqrt[3]{-9}$ .

What you need and how to start

What you need: $a<b\iff\sqrt[3]a<\sqrt[3]b$ .

Steps:

Write $5=\sqrt[3]{125}$ and $-2=\sqrt[3]{-8}$ .
Compare the numbers under the root signs.

Show solution

a) We have $5=\sqrt[3]{125}$ . Since $125>124$ , $5>\sqrt[3]{124}$ . b) We have $-2=\sqrt[3]{-8}$ . Since $-8>-9$ , $-2>\sqrt[3]{-9}$ .

Answer. a) $5>\sqrt[3]{124}$ ; b) $-2>\sqrt[3]{-9}$ .

Exercise 5

Find $x$ given that: a) $x^3=-64$ ; b) $\sqrt[3]{3x+1}=-2$ .

What you need and how to start

What you need: $x^3=a\iff x=\sqrt[3]a$ ; $\sqrt[3]A=b\iff A=b^3$ .

Steps:

Part a: take the cube root.
Part b: cube both sides and solve the linear equation.

Show solution

a) We have $x^3=-64\iff x=\sqrt[3]{-64}=-4$ . b) We have $\sqrt[3]{3x+1}=-2\iff 3x+1=(-2)^3=-8\iff 3x=-9\iff x=-3$ .

Answer. a) $x=-4$ ; b) $x=-3$ .

Exercise 6

Simplify the following expressions: a) $\sqrt[3]{27x^3}-2x$ ; b) $\sqrt[3]{x^3+6x^2+12x+8}$ .

What you need and how to start

What you need: $\sqrt[3]{A^3}=A$ ; the identity $(a+b)^3=a^3+3a^2b+3ab^2+b^3$ .

Steps:

Write $27x^3=(3x)^3$ .
Recognise $x^3+6x^2+12x+8=(x+2)^3$ .

Show solution

a) We have $\sqrt[3]{27x^3}-2x=\sqrt[3]{(3x)^3}-2x=3x-2x=x$ . b) We have $\sqrt[3]{x^3+6x^2+12x+8}=\sqrt[3]{(x+2)^3}=x+2$ .

Answer. a) $x$ ; b) $x+2$ .

Exercise 7

A cube-shaped water tank holds $3375$ litres when full.

a) Find the edge of the tank in metres.

b) Find the area of the base of the tank.

What you need and how to start

What you need: $1$ litre $=1$ dm³; the edge of a cube is $\sqrt[3]{V}$ .

Steps:

Work out $\sqrt[3]{3375}$ (dm) and convert to metres.
The base area is the square of the edge.

Show solution

a) We have $3375$ litres $=3375$ dm³. Hence the edge of the tank is $\sqrt[3]{3375}=15$ (dm) $=1.5$ (m).

b) The area of the base is $1.5^2=2.25$ (m²).

Answer. a) $1.5$ m; b) $2.25$ m².

Exercise 8

A large cube is built from $512$ identical small cubes, and then its whole outside surface is painted.

a) How many small cubes are there along each edge of the large cube?

b) How many small cubes have no painted face?

What you need and how to start

What you need: the number of cubes along an edge is $\sqrt[3]{512}$ .

Steps:

Work out $\sqrt[3]{512}$ .
The unpainted cubes form a smaller cube inside, with $2$ fewer cubes along each edge.

Show solution

a) The number of small cubes along each edge is $\sqrt[3]{512}=8$ .

b) The unpainted small cubes form a cube inside, with $8-2=6$ cubes along each edge. Hence $6^3=216$ small cubes have no painted face.

Answer. a) $8$ ; b) $216$ .

Exercise 9

Evaluate $\sqrt{16}+\sqrt[3]{-27}+\sqrt{(-2)^2}$ .

What you need and how to start

What you need: $\sqrt{a^2}=|a|$ (see square roots and square-root expressions) and $\sqrt[3]{a^3}=a$ .

Steps:

Work out each term, watching the signs.
Add them.

Show solution

We have $\sqrt{16}+\sqrt[3]{-27}+\sqrt{(-2)^2}=4+(-3)+|-2|=4-3+2=3$ .

Answer. $3$ .

Exercise 10 Challenge

How many integers $x$ satisfy $-2<\sqrt[3]{x}<3$ ?

What you need and how to start

What you need: $a<b\iff\sqrt[3]a<\sqrt[3]b$ , so we can cube every part and keep the inequality signs.

Steps:

Write $-2=\sqrt[3]{-8}$ and $3=\sqrt[3]{27}$ .
Count the integers in the interval you get.

Show solution

We have $-2=\sqrt[3]{-8}$ and $3=\sqrt[3]{27}$ . Hence $-2<\sqrt[3]{x}<3\iff -8<x<27$ . So the integers $x$ are $-7, -6, \ldots, 26$ , which is $26-(-7)+1=34$ numbers.

Answer. $34$ integers.

a) $A$ is defined if and only if $x\ge7$ .	True	False
b) When $x=-1$ , $A=-2$ .	True	False
c) $A=3$ when $x=34$ .	True	False
d) $A<0$ when $x<7$ .	True	False

The cube root of a real number

Explore: the cubes of a few numbers

Comparing cube roots

Cube-root expressions

Key terms

Summary

Practice

Mini test

Part I. Multiple choice

Part II. True or false

Part III. Short answer