The a3-b3formula is called the difference of cubes (of two numbers) formula. The a cube minus b cube formula isused to find the difference between the two cubes without actually calculating the cubes. Also, it is used to factorize the binomials of cubes.In this section, we will be discussing the various aspects of the a3-b3formula, along with solved examples, and understand the identityinvolved.

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What Is a^3-b^3 Formula?

Thea3-b3formula can beverified, bymultiplying(a - b) (a2+ab + b2) and see whether you get a3- b3.The a3- b3formula or the difference ofcubesformula is explained below:

a3-b3Formula = a3- b3= (a -b) (a2+ ab + b2)

You can remember these signs using the following trick.

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Let us learn thea3-b3formula with a few solved examples.

Proof of a3-b3 Formula

Let us verify the a cube minus b cube formula. To provethata3- b3= (a -b) (a2+ ab + b2) we need to prove here LHS = RHS. Lets begin with the following steps.LHS =a3- b3On Solving RHS side we get,= (a -b) (a2+ ab + b2)On multiplying the a and b separatelywith(a2+ ab + b2) we get= a (a2+ ab + b2) - b(a2+ ab + b2)= a3 + a2b + ab2 - a2b - ab2- b3= a3 + a2b - a2b + ab2- ab2- b3=a3- 0 - 0 - b3=a3- b3Hence proved, LHS = RHS



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Examples on a^3-b^3Formula

Example 1:Find the value of 1083-83using the a^3-b^3formula.

Solution:

To find:1083-83.

Let us assume that a = 108and b = 8.

We will substitute these in the formula ofa3- b3.

a3- b3= (a -b) (a2+ ab + b2)

1083-83= (108-8)(1082+(108)(8) + 82)

= (100) (11664+864+64)

= (100)(12592)

=1259200

Answer:1083-83=1,259,200.

Example 2:Factorize the expression 27x3-125usinga^3-b^3 formula.

Solution:

To factorize: 27x3- 125.

We will use thea3- b3formula to factorize this.

We can write the given expression as

27x3- 125 = (3x)3- 53

We will substitute a = 3x and b = 5in the formula ofa3- b3.

a3- b3= (a -b) (a2+ ab + b2)

(3x)3-53=(3x-5)((3x)2+(3x)(5)+52)

= (3x-5) (9x2+15x+25)

Answer:27x3-125= (3x -5) (9x2+ 15x + 25).

Example 3: Simplify 193 - 203 using a cube minusb cubeformula.

Solution: To find193 - 203

Let us assume a = 19 and b = 20Using formulaa3- b3= (a - b) (a2+ ab + b2)We will substitute these in thea3- b3formulaa3- b3= (a -b) (a2+ ab + b2)193-203= (19-20)(192+(19)(20)+202)= (-1)(361+380+400)= (-1)(1141)= -1141Answer:193-203= -1141.


FAQ's on a^3-b^3Formula

What Is the Expansion of a3-b3 Formula?

a3-b3formula is read as a cube minus b cube. Its expansion is expressed asa3-b3=(a -b) (a2+ ab + b2).

What Is thea3-b3 Formula in Algebra?

The a3-b3formula is also known as one of the important algebraic identiy. It is read as a cube minus b cube. Its a3-b3formula is expressed asa3-b3=(a -b) (a2+ ab + b2).

How To Simplify Numbers Usingthea3-b3 Formula?

Let us understand the use of the a3-b3formula with the help of the following example.Example: Find the value of 103- 23using the a3-b3formula.To find:103-23Let us assume that a = 10and b = 2.We will substitute these in the formula ofa3- b3.a3- b3= (a -b) (a2+ ab + b2)103-23= (10-2)(102+(10)(2)+22)= (8) (100+20+4)= (8)(124)=992Answer:103-23= 992.

How To Use thea3-b3 Formula Give Steps?

The following steps are followed while usinga3-b3formula.

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Firstlyobserve the pattern of the numbers whether thenumbers have ^3 as power or not.Write down the formula ofa3- b3 a3- b3= (a -b) (a2+ ab + b2)substitute the values ofa and b in thea3-b3formula and simplify.