Cubes and Cube Roots

In mathematics, we use the 'cube' word in two places. First, to describe a three-dimensional figure and second, while dealing with exponents, i.e. \(x^3\) or x-cubed.

As a three-dimensional object, a cube looks like this.

cube

Are both the interpretations of the word 'cube' related?

Let's find out.

In this mini-lesson, we will explore cubes and cube roots by learning about their definitions, how to find cubes and cube roots, and about perfect cubes through interesting simulations and visualization, some solved examples, and a few interactive questions for you to test your understanding.

Lesson Plan

What Do you Mean By Cubes?

When we multiply a number three times to itself, the resultant number is known as the cube of the original number.

We call it a cube because it is used to represent the volume of a cube.

In other words, a number raised to exponent 3 is called a cube of that number. We read it as 'number-cubed' like, 1-cubed, 2-cubed, 3-cubed, etc.

\(x=y^3\)

Where \(x\) is the cube of \(y\). 

For example, a cube of number 3 or 3 cubed is 27

\[\begin{align} 3\times 3 \times 3=27 \end{align}\]

The cube of number 4 or 4 cubed is 64

\[\begin{align} 4\times 4 \times 4=64 \end{align}\]

The cube of number 5 or 5 cubed is 125

\[\begin{align} 5\times 5 \times 5=125 \end{align}\]


How Do you Find Cube of Any Number?

To find the cube of a number, first, multiply that number by itself, then multiply the product obtained with the original number again.

Let us find the cube of \(7\) through the same process.

\(7\times 7 \times 7\)

Step 1- Multiply 7 with itself.

            \(7 \times 7=49\)

Step 2- Multiply \(49\) by \(7\) again.

            \(49\times 7=343\)

\(\therefore\) \(7\) cubed is \((7)^3\), which is \(343\).

Cube of a fraction

For finding a cube of a fraction, first, we write the multiplication of that number three times by itself. Then, we multiply the numerators and denominators separately to find the answer.

Let us find the cube of number \(\dfrac{2}{5}\).

Cube of \(\dfrac{2}{5}\) is \((\dfrac{2}{5})^3\).

Step 1- \(\dfrac{2}{5}\times \dfrac{2}{5}\times \dfrac{2}{5}\).

Step 2- Multiply the numerators and denominators separately.

            \(\dfrac{2\times 2\times 2}{5\times 5\times 5}=\dfrac{8}{125}\)

Cube of Negative Numbers

The process to find the cube of a negative number is the same as that of a whole number and fraction.

Here, always remember that the cube of a negative number is always negative, while the cube of a positive number is always positive.

Let us find the cube of \(-6\)

Cube of \(-6\) is \((-6)^3\)

Step 1- \((-6) \times (-6)\times (-6)\) 

            Multiply \(-6\) with \(-6\), we get \((-6)\times (-6)=36\)

Step 2- Multiply \(36\) again with \(-6\)

            \(36\times (-6)=-216\)

\(\therefore (-216)\) is the cube of \(-6\)


What do you Mean by Cube Roots?

When we think about the words cube and root, the first picture that might come to our mind is:

cube

Isn't it?

Well, the idea is similar. Root means the primary source or origin.

So, we just need to think "cube of which number should be taken to get the given number".

In mathematics, cube roots definition is "Cube root is the number that needs to be multiplied three times to itself to get the original number."

Now, let us look at the cube root formula, where \(y\) is the cube root of \(x\).

\(\sqrt[3]{x}=y\)

Radical sign \((\sqrt{})\) is used as a cube root symbol for any number with a small 3 written on the left of the sign. Another way to denote cube root is to write \(\frac{1}{3}\) as the exponent of a number.

It is an inverse operation of the cube of a number.

cube root as inverse operation of cube

Let's look at some cube roots examples. If we take 3 cubed, we get 27. So, 3 is the cube root of 27

Similarly, if we take 4 cubed, we get 64. So, 4 is the cube root of 64

cube and cube root

If the cube root of a number is a whole number, we call that number a perfect cube.

Some examples of perfect cubes are, \(1\), \(8\), \(27\), \(64\), etc.

It means that the cube of a whole number is always a perfect cube.

Look at the perfect cubes table given below showing perfect cubes of the first 10 natural numbers.

Number Perfect cube
1 1
2 8
3 27
4 64
5 125
6 216
7 343
8 512
9 729
10 1000
 
important notes to remember
Important Notes
  1. 1 is the cube and cube root of itself, as, \(1\times 1\times 1=1\) and \(\sqrt[3]{1}=1\), also satisfying the cube roots definition. 
  2. The unit place of a number can be identified by looking at the unit place of its cube. For example, 6859 is the cube of a number whose unit place is 9 as there is no other number whose cube has 9 digits as the unit place.
  3. Radical sign(square root symbol) is used to denote cube root with a small 3 written on the left side of the sign \((\sqrt[3]{})\).

How Do You Find Cube Root of Any Number?

The cube root of a number can be determined by the use of the prime factorization method

Steps to be followed to find the cube root of a number using the prime factorization method is written below:

Step 1- Do the prime factorization of the given number.

Step 2- Divide the factors obtained into three groups containing the same number of each factor.

Step 3- Multiply the factors in any one of the groups. That number will be your answer.

If there is any factor left that cannot be divided equally into three groups, that means the given number is not a perfect cube.

To find another trick of finding whether a number is a perfect cube or not, refer to the tips and tricks section given below.

Let's understand it better with the help of a cube roots example.

Example and Solution

Find the cube root of \(1728\)

Step 1- First, we need to do the prime factorization of the given number as follows.

prime factorization of 1728

Step 2- Now, let's make 3 groups of the same numbers.

make groups of common factors

Step 3- Multiply the common factors.

\[\begin{align} 2\times 2\times 3=12 \end{align}\]

\(\therefore 12\) is the cube root of \(1728\)

Cube Root of a Fraction

It is very easy to find the cube root of a fraction. We just have to find the cube root of the numerator and denominator separately.

For example, cube root of \(\dfrac{8}{27}\) is \(\dfrac{2}{3}\), as the cube roots of \(8\) and \(27\) are \(2\) and \(3\) respectively.

Cube Root of Negative Numbers

The only difference between the cube root of a negative number and the cube root of a positive number is the presence of a negative sign with the cube root of the negative number.

Some examples are shown below:

\[\begin{align} \sqrt[3]{-343}=-7 \end{align}\]

\[\begin{align} \sqrt[3]{-\dfrac{27}{64}}=-\dfrac{3}{4} \end{align}\]

Cube Root Calculator

In the simulation given below of cube root calculator, write the number whose cube root you want to find and click on 'Calculate'.

 
tips and tricks
Tips and Tricks
  1. Let us learn a trick to find whether a number is a perfect cube or not.

Look at the cube roots table given below and observe the pattern in the sum of the digits of perfect cubes.

Cube root Cube Sum of digits in a cube
2 8 8
3 27 2+7=9
5 125 1+2+5=8
8 512 5+1+2=8
9 729

7+2+9=18=1+8=9

10 1000 1
12 1728 1+7+2+8=18=1+8=9
13 2197 2+1+9+7=19=1+9=10=1+0=1
20 8000 8
Yes, the sum of the digits of a perfect cube number is always 1, 8, or 9.
Note: The converse of this is not true. Every number whose sum of digits is 1, 8, or 9 need not be a perfect cube.

Solved Examples

Example 1

 

 

Emily's father's age is 27 years. Find the age of Emily if her age is the cube root of her father's age.

Solution

Given, age of Emily's father= 27 years.

Emily's age= cube root of 27

                  = \(\sqrt[3]{27}\) years

prime factorization of 27

\(\therefore\) Emily is 3 years old.
Example 2

 

 

Walter was playing with the Rubik's cube and noticed that it is actually a perfect cube.

He asks "How to find the volume of a Cube?"

Can you help him?

rubik cube

Solution

Length of the cube (s) = 4 inches

Since, volume of the cube = \( \text s^3 \)

where \('s'\) is the length of the cube.

Then,  

Volume of the cube = \((4^3)\;\text{in}^3 = 64\;\text{in}^3 \)

\( \therefore\) Volume of the Rubik's cube is \(64\ {in}^3\)
Example 3

 

 

Two friends Jimmy and Sam are arguing with each other. Let's see what they are talking about.

Jimmy- "Like square of a number, cubes of number \(-3\) and \(3\) are same."

Sam- "No, cubes of \(-3\) and \(3\) are different."

two kids arguing about cube roots

Can you tell who is right and why?

Solution

Sam is right as unlike the square of a number, cubes of a number depend on whether the number is positive or negative.

Cubes of positive numbers are always positive and cubes of negative numbers are always negative.

So, 

\(3^3=3\times 3\times 3=27\)

\((-3)^3=(-3)\times (-3)\times (-3)=-27\)

\(\therefore\) Sam is right.

Interactive Questions

Here are a few activities for you to practice. Select/Type your answer and click the "Check Answer" button to see the result.

 
 
 
 
 
 

Let's Summarize

The mini-lesson targeted the fascinating concept of cubes and cube roots. The math journey around cubes and cube roots started with what a student already knew and went on to creatively crafting a fresh concept in the young minds. Done in a way that not only it is relatable and easy to grasp, but also will stay with them forever.

About Cuemath

At Cuemath, our team of math experts is dedicated to making learning fun for our favorite readers, the students!

Through an interactive and engaging learning-teaching-learning approach, the teachers explore all angles of a topic.

Be it worksheets, online classes, doubt sessions, or any other form of relation, it’s the logical thinking and smart learning approach that we, at Cuemath, believe in.


FAQs on Cubes and Cube Roots 

1. What are cubes and cube roots?

The cube of a number is the value of the third exponent of the number. For example, \(2^3\).

While the cube root of a number is the number that needs to be multiplied by itself 3 times to get the original number. For example, 2 is the cube root of 8

2. What are cube roots used for?

Cube roots are used to solve cubic equations and also to find the dimension of a cube if the volume is given.

3. How do you simplify cube roots?

Cube roots can be simplified using the prime factorization method. First, do the prime factorization, then take out common factors in groups of 3. Multiply those common factors to get the answer.

4. Is it possible to simplify the negative cube root?

Yes, the simplification of negative cube roots is the same as positive cube roots. The only difference is the presence of a negative sign with the cube root of a negative number.

5. What is not a perfect cube?

A number is not a perfect cube if we cannot make 3 equal groups of factors of the number after doing prime factorization of the number.

6. What is the cube of an odd natural number?

The cube of an odd natural number is always an odd number. For example, \(5^3=125\),  \(7^3=343\),  \(9^3=729\), etc.

7. Can the cube root of any odd number be even?

No, the cube root of an odd number is always odd.

8. What is an easy way to calculate the cube root of any number?

The easiest and basic method to find the cube root of any number is the prime factorization method.

9. What is the way to find cube roots of non-perfect cubes?

To find the cube root of a non-perfect cube, say, 265, follow the steps given below:

Step 1- Find an integral part by taking the smaller value out of the cube roots of two nearby perfect cubes.

            Two perfect cubes nearby 265 are 216 and 343. Out of these two numbers, the cube root of the smaller number, i.e. 216 is 6

Step 2- Divide the number by the square of the value of the cube root found in step1.

            \(6^2=36\)

            \(265 \div 36=7.36\)

Step 3- Subtract 6 from 7.36 and divide the number so obtained by 3

            \(7.36-6=1.36\)

            \(1.36\div 3=0.45\)

Step 4- Add the integral cube root found in step1 and the number found in step 3

            \(6+0.45=6.45\)

\(\therefore 6.45\) is the approximate value of the cube root of 265, which is a non-perfect cube.

10. What is the formula of the cube root?

The formula of cube root is \(a=\sqrt[3]{b}\), where a is the cube root of b.
  
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