# 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. 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

 1 What Do you Mean By Cubes? 2 Important Notes 3 Tips and Tricks 4 Solved Examples 5 Interactive Questions

## 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: 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 $$x$$.

 $$\sqrt{x}=y$$

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. 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 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

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