LemonmathSelf-study sheet

Grade 9 · Lesson 3.4

Cube roots and cube-root expressions

Name: Date: Time spent: min

Goals

Understand the cube root of a real number and find the cube roots of simple numbers.
Use a calculator to find approximate cube roots.
Recognise cube-root expressions and use $\sqrt[3]{A^3}=A$ to evaluate and simplify.

Remember

$x=\sqrt[3]{a}\iff x^3=a$ . 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 $\sqrt[3]{V}$ .

Exercises

Exercise 1

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

Exercise 2

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

Exercise 3

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

Exercise 4

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

Exercise 5

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

Exercise 6

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

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.

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?

Exercise 9

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

Exercise 10 Challenge

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

Mini test

15 minutes, without looking back at the lesson. Part I: circle one letter; Part II: write T or F for each statement; Part III: write the answer in the box.

Part I. Multiple choice

Question 1.

\sqrt[3]{-27}

equals

-3

.B.

3

.C.

-9

.D. not defined.

Question 2. Which expression is defined for every real number

x

\sqrt{x-1}

.B.

\sqrt[3]{x-1}

.C.

\sqrt{1-x}

.D.

\sqrt{x}

Question 3.

\sqrt[3]{(-5)^3}

equals

5

.B.

-5

.C.

125

.D.

\pm5

Question 4. A cube-shaped crate has a volume of

0.216

m³. Its edge is

0.6

m.B.

0.072

m.C.

0.06

m.D.

6

Part II. True or false

Question 1. Consider the expression

A=\sqrt[3]{x-7}

. Decide whether each statement is true or false.

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

Part III. Short answer

Question 1. Evaluate

\sqrt[3]{125}-\sqrt[3]{-8}+\sqrt[3]{0.001}

Question 2. A cube-shaped gift box has a volume of

3

litres. Find the length of its edge in centimetres (give your answer to 1 decimal place).

Score: ........ / 10

Hints (only when stuck)

Exercise 1.

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.

Exercise 2.

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

Steps:

Work out each root.
Add and subtract.

Exercise 3.

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.

Exercise 4.

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.

Exercise 5.

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.

Exercise 6.

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

Exercise 7.

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.

Exercise 8.

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.

Exercise 9.

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.

Exercise 10.

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.

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