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Cube Root Of Negative Number

Cube Numbers And Cube Roots

Hither nosotros will acquire about cube numbers and cube roots including what a cube number is, what a cube root is, as well as how to cube a number and how to find the cube root of an integer. You'll too learn how to solve bug by applying cognition of cube numbers.

There are also cube numbers and cube roots worksheets based on Edexcel, AQA and OCR examination questions, forth with further guidance on where to become adjacent if you're yet stuck.

What is a cube number?

A cube number is found when we multiply an integer (whole number) by itself and and so itself once again. These are sometimes called 'perfect cubes'.
We tin cube numbers with decimals places but nosotros practise not refer to these as cube numbers.
A number/variable that is 'cubed' is multiplied by itself three times.

E.g.
four × 4 × 4 tin be written every bit 43 and is spoken equally "iv cubed " or "four to the power of iii"

A cube number is found when we multiply an integer (whole number) by itself,

Hither are the first 10 cube numbers:

\brainstorm{align*} 1 \times 1 \times 1 &=ane \quad \quad \quad \quad \quad \quad \quad \quad \quad \text { Therefore } i \text { is a cube number } \\\\ 2 \times two \times 2&=viii \quad \quad \quad \quad \quad \quad \quad \quad \quad \text { Therefore } 8 \text { is a cube number } \\\\ 3 \times three \times 3&=27 \; \; \quad \quad \quad \quad \quad \quad \quad \quad \text { Therefore } 27 \text { is a cube number } \\\\ 4 \times four \times 4&=64 \; \; \quad \quad \quad \quad \quad \quad \quad \quad \text { Therefore } 64 \text { is a cube number } \\\\ v \times 5 \times five&=125 \; \quad \quad \quad \quad \quad \quad \quad \quad \text { Therefore } 125 \text { is a cube number } \\\\ half dozen \times half-dozen \times 6&=216 \; \quad \quad \quad \quad \quad \quad \quad \quad \text { Therefore } 216 \text { is a cube number } \\\\ vii \times 7 \times 7&=343 \; \quad \quad \quad \quad \quad \quad \quad \quad \text { Therefore } 343 \text { is a cube number } \\\\ 8 \times 8 \times 8&=512 \; \quad \quad \quad \quad \quad \quad \quad \quad \text { Therefore } 512 \text { is a cube number } \\\\ 9 \times 9 \times 9&=729 \; \quad \quad \quad \quad \quad \quad \quad \quad \text { Therefore } 729 \text { is a cube number } \\\\ ten \times x \times 10&=yard \quad \quad \quad \quad \quad \quad \quad \quad \text { Therefore } m \text { is a cube number } \end{align*}

A cube number can be represented every bit an array which forms the shape of a cube that has a length 3 units, width 3 units and depth iii units.

East.g.

If we look at three3 , this is 3 × iii × 3

Any cube number will form the shape of a cube.

What is a cube number?

Cubing negative numbers

We can also cube negative numbers.
E.one thousand.

\brainstorm{assortment}{ll} (-v) \times(-five) \times(-5)=-125 & \quad \therefore(-5)^{three}=-125 \\\\ (-7) \times(-7) \times(-7)=-343 & \quad \therefore(-7)^{3}=-343 \end{array}

Y'all will notice that when we cube a negative number we get a negative number.
This is considering a negative number multiplied by a negative number multiplied by a negative number gives us a negative upshot.
Acquire more by reviewing our lesson on negative numbers.

When we cube negative 5 nosotros get the negative reply of the cube of positive 5.
This is true for all numbers (and variables) and means:

\[(5)^{three} \neq(-v)^{3}\]

What is a cube root?

The cube root of a number is a value that can be multiplied by itself three times to requite the original number.
A cube root is the inverse functioning of cubing a number.
The cube root function looks similar this iii

When we cube a positive number we get a positive effect and when nosotros cube a negative number we become a negative event.
And so the cube root of a positive number is besides a positive number, and the cube root of a negative number is besides a negative number.

Due east.g.

\begin{array}{50} \text { As } 3^{iii}=27 \text { the cube root of } 27 \text { is } three & \quad \therefore \sqrt[iii]{27}=3 \\\\ \text { As } 8^{iii}=512 \text { the cube root of } 512 \text { is } eight & \quad \therefore \sqrt[3]{512}=8 \\\\ \text { Every bit} (-3)^{3}= -27 \text { the cube root of} -27 \text { is} -3 & \quad \therefore \sqrt[iii]{-27}=-3 \cease{array}

What is a cube root?

What is a cube root?

Cube numbers worksheet

Cube numbers worksheet

Cube numbers worksheet

Go your complimentary cube numbers worksheet of 20+ questions and answers. Includes reasoning and applied questions.

DOWNLOAD FREE

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Cube numbers worksheet

Cube numbers worksheet

Cube numbers worksheet

Get your free cube numbers worksheet of twenty+ questions and answers. Includes reasoning and applied questions.

DOWNLOAD FREE

Key words

Term

A single number (constant) or variable

Due east.g.
in the expression 4x − 7 both 4x and − 7 are terms


Coefficient

The number which the variable is being multiplied by

E.g.
in 2xiii the coefficient is 2


Integer

A whole number

E.thou.
1, 7 or 1003


Index (also called exponent or powers)

The index number is the corporeality of times you multiply a number/variable by itself.

Due east.g.
the index number in 5iii is iii

Annotation: the plural of alphabetize is indices

Note: yous will encounter index number as a superscript


Base number

The number/unknown that is beingness multiplied by itself an amount of times

E.g.
the base number in 5three is 5 and in 2x3 the base number is 10

Advanced vocabulary –  only for Additional Maths, A-Level

Real numbers

Whatsoever positive or negative number is chosen a real number. Numbers that are not 'existent' are called imaginary numbers. Integers, decimals, fractions are all examples of existent numbers

E.g.
i, two, 5 and 100 are all examples of a positive existent numbers

-i, -2, -50 and -65.67 are all examples of a negative real number


Imaginary numbers

Numbers that are not existent are chosen imaginary numbers, for example you will notice we cannot find the square root of a negative number (endeavor it on a calculator), this considering it is an imaginary number.. Numbers that contain an imaginary office and real part are chosen circuitous numbers.

Cube numbers and cube roots examples

Example 1

What is v cubed?

5 cubed means 5 × five × 5

So 5 cubed is 125

Example 2

What is nine cubed?

nineiii cubed ways nine × nine × 9

So 9 cubed is 729

Example 3

What is the cube root of 64?

The cube root of 64 means what value tin can be multiplied by itself three times to requite 64

So

\[\sqrt[three]{64} =four\]

How to use cube numbers and cube roots

  1. Place whether you need to cube or cube root the number/variable
  2. Perform the operation
  3. Clearly state the reply within the context of the question e.k. including units

Now yous volition focus on solving problems using your knowledge of cube numbers and cube roots.

Explain how to solve problems involving cube numbers and cube roots in 3 steps

Explain how to solve problems involving cube numbers and cube roots in 3 steps

Cube numbers and cube roots problem examples

Example four: noesis of cube numbers

Danny says ii cubed is 6. Why is Danny wrong? What mistake did he brand?

  1. Identify whether yous demand to cube or cube root the number/variable

The question focus is on cubing ii or "2 cubed"

two Perform the operation

\brainstorm{array}{fifty} ii \times two \times ii =2^{3} =eight \finish{array}

3 Clearly state the respond within the context of the question

Danny is wrong because 2 cubed is 8 non half dozen.

The mistake Danny made was he did '2 × 3' not '23'.

Example 5: problem solving with cube roots

Ava says the cube root of an integer is always smaller than the original number. Testify Ava is wrong.

You are looking for a relationship between the cube root of an integer and the original number. It will aid to write down your cube numbers and their cube roots.

\brainstorm{array}{l} \sqrt[iii]{1000}=10 \\\\ \sqrt[3]{729}=ix \\\\ \sqrt[3]{512}=eight \\\\ \sqrt[3]{343}=7 \\\\ \sqrt[three]{216}=half dozen \\\\ \sqrt[3]{125}=5 \\\\ \sqrt[three]{64}=4 \\\\ \sqrt[3]{27}=3 \\\\ \sqrt[iii]{eight}=2 \\\\ \sqrt[3]{1}=ane \end{array}

\[\sqrt{1}=1\]


The respond here is not smaller than the original number.

Example half-dozen: solving problems involving cube numbers

The sum of 2 cube numbers is 72. Notice the two cube numbers.

Remember sum means add. Therefore yous are looking for two cube numbers that add together to make 72.


It will assist here to list the cube numbers up to 72

The Cube Numbers: 1     8     27     64


You lot at present need to pick two of these numbers that when added together make 72

The ii cube numbers are 64 and 8.

Instance 7: cube numbers within a 3D polygon

A cube has a volume of 125mm 3 . What is the length of ane side?

The volume of a cube is found past multiplying the length, width and peak together. For a cube the length, width and tiptop are the aforementioned length.


Therefore yous are looking for a number that when multiplied past itself three times (or cubed) is equal to 125.


Therefore yous need to find the cube root of 125

Therefore the length of the cube is vmm.

Instance eight: cube numbers within a 3D polygon

Length of ane side of a cube is 4mm. What is the volume of the cube?

The book of a cube is institute past multiplying the length, width and top together. For a cube the length, width and acme are the aforementioned length.


Therefore you are are going to multiply the side length by itself three or 'cube' the length.


Therefore you need to cube 4

Therefore the volume of the cube is 64mm 3 .

Example nine

Lexi says "when yous add three consecutive cube numbers, the answer is always odd."

Is Lexi correct? Explain your respond.

To prove Lexi wrong we simply need to discover one example where she is incorrect, this is sometimes known as proof past contradiction.


Therefore we are looking for 3 sequent cube numbers that when added together are even. Information technology will exist helpful here to list the 'principal' cube numbers.

\[1, \quad 8, \quad 27, \quad 64, \quad 125, \quad 216, \quad 343, \quad 512, \quad 729, \quad yard \]


You are now looking for one example where three of these numbers when added together make an even number.


For case: one + viii + 27 = 36

Lexi is wrong because i + 8 + 27 = 36

Common misconceptions

  • Cube numbers

Incorrect understanding of cubing a number

Eastward.thou.

iii3 = 27

non

nine
  • Cube roots

Not recognising that a negative number cube rooted is negative

E.g.
\sqrt[3]-viii = -2

Practice cube numbers and cube roots questions

GCSE Quiz False

GCSE Quiz False

GCSE Quiz True

GCSE Quiz False

10^{3}=10 \times 10 \times x = 1000

GCSE Quiz False

GCSE Quiz True

GCSE Quiz False

GCSE Quiz False

six \times 6 \times 6 = 216 \quad \text{ therefore } \quad \sqrt[three]{216} = vi

GCSE Quiz False

GCSE Quiz True

GCSE Quiz False

GCSE Quiz False

(-7)^{3} = (-7) \times (-7) \times (-7) = -343

GCSE Quiz False

GCSE Quiz False

GCSE Quiz False

GCSE Quiz True

(-1) \times (-1) \times (-1) = -1 \quad \text{ therefore } \quad \sqrt[3]{-one}=-one

GCSE Quiz False

GCSE Quiz False

GCSE Quiz True

GCSE Quiz False

The cube numbers up to 65 are ane, 8, 27, 64.

The 2 that add up to 65 are 1 and 64.

GCSE Quiz False

GCSE Quiz True

GCSE Quiz False

GCSE Quiz False

Nosotros find the book of a cube by multiplying the length, the width and the superlative together.

These are all equal for a cube and so we need to find a number that, when multiplied by itself 3 times, gives us 512.

Therefore we need to find the cube root of 512.

\sqrt[3]{512}=8

Cube numbers and cube roots GCSE questions

1. Work out the value of:

(a) 3^{3}

(b) five \times two^{three}

(c) 6^{3} − iii^{3}

(3 Marks)

Testify reply

(a) 3 \times iii \times 3

27

(one)

(b) five \times viii

40

(ane)

(c) 216 – 27

189

(i)

2. Here is a list of numbers:

1000 \quad \quad 18 \quad \quad 8 \quad \quad 64 \quad \quad seven \quad \quad 144 \quad \quad one \quad \quad xix

(a)List the cube numbers

(b)Which number is a square and cube number?

(2 Marks)

Bear witness answer

(a) k, 8 , 64 and 1

(one)

(b) 64

(1)

three. Find the value of:

(a) \sqrt[3]{729}

(b) \sqrt[3]{-1}

(c) \sqrt[iii]{-8}

(3 Marks)

Bear witness answer

(a) nine

(1)

(b) -1

(1)

(c) -two

(1)

iv. Simplify the following expression

5^{3} \times \sqrt[three]{x^{3}}

(2 Marks)

Show answer

125 10

Correct coefficient

(1)

Correct x (or x^{1} )

(ane)

Learning checklist

You have now learned how to:

  • Calculate cube numbers up to x × x × 10
  • Use positive integer powers and their associated real roots
  • Recognise and use the cube numbers
  • Apply properties of cubes to a context

Still stuck?

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Cube Root Of Negative Number,

Source: https://thirdspacelearning.com/gcse-maths/number/cube-numbers/

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