Cube of 12- Value and methods to find it

Multiplying a number three times by itself yields a number which is the cube of the original number. The exponential form of a cube is x³. So, In this article, we will discuss the value of the cube of 12 and the methods to find the cube.

Cube of 12

The value of the cube of 12 is 1728. We will find the cube of 12 by using the multiplication method and algebraic expressions.

Method 1 – Multiplication method

In this method, we need to multiply the number three times by itself to get the cube of that number. It is the simplest method to find the cube of any small number.

The cube of 12 = 12³

= 12×12×12

= 1728

Method 2 – Algebraic expression

In this method, we can use either of the two algebraic expressions to find the cube. We will get the same answer from both expressions. The expressions are –

Expression 1- (p + q)³ = p³ + q³ + 3pq(p + q)

Expression 2- (p – q)³ = p³ – q³ – 3pq(p – q)

We will find the cube of 12 by using both the expressions –

First, let us write 12 as (10 + 2) for using the first expression, where p = 10, q = 2

(10 + 2)³ = 10³ + 2³ + 3×10×2(10 + 2)

= 1000 + 8 + 60×12

= 1728

Now, we will use the second expression. For that, we will write 12 as (15 – 3), where p=15, q=3.

(15 – 3)³ = 15³ – 3³ – 3153(15 – 3)

= 3375 – 27 – 135×12

= 3375 – 1647

= 1728

Illustration

Q1. Find the value of a and b when a³ × b³ = 12³.

Sol. We know that the value of the cube of 12 = 1728

To find the value of a and b, we need to do the prime factorisation of 1728.

The prime factorisation of 1728 = 2×2×2×2×2×2×3×3×3

We will make the pair of threes – 2³×2³×3³

You can see that we have to find the value of a and b, but we are getting three pairs.

We will use the identity – x^m×y^m = (xy)^m.

Using the above formula we can write 1728 = (2×2)³ × 3³.

The value of a³ × b³ = 4³×3³,

if a=4, then b=3

Or    if a=3, then b=4.

We can also write 1728 = (2×3)³ × 2³= 6³×2³.

Here, a³ × b³ = 6³×2³, if a=6, then b=2 or if a=2, then b=6.

Q2. When the cube of 12 is divided by the cube of 6, we get the cube of which number?

Ans. The values of the cube of 6 and 12 are 216 and 1728 respectively.

When 1728 is divided by 216 = 8.

The prime factors of 8 are 2×2×2 = 2³.

On applying the cube root on 2³, we get 2 as the cube root of 8.

Therefore, when the cube of 12 is divided by the cube of 6, we get the cube of 2.