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# A cube plus B cube formula ## What is the formula of A cube plus B cube

A cube plus B cube formula is an algebraic identity which is used to find the sum of the two cubes. The expression of this A cube plus B cube formula is a+ b= (a + b) (a2 – ab + b2)

### How to derive A cube plus B cube ka formula

Lets understand this a+ b3 Formula in details. this formula is the sum of the two cubes. Here we will see the other expressions of A cube plus B cube ka formula or you can say other forms of a+ b3 formula.

We know already the formula of (a+b)3 Here we will use this A plus B ka whole cube formula to prove A cube plus B cube ka formula. so (a+b)3=a3+b3+3ab(a+b).

(a+b)3=a3+b3+3ab(a+b)

a3+b3 = (a+b)3–3ab(a+b)

a3+b3=(a+b)3–3ab(a+b) [Note: Take comman part (a+b)]

a3+b3=(a+b)((a+b)2–3ab)

We know that formula of A plus B ka whole square (a+b)2 = a2+b2+2ab

a3+b3=(a+b)(a2+b2+2ab–3ab)

a3+b3=(a+b)(a2+b2–ab)

### Proof of a3+ b3 Formula

Now we will prove whether this A cube plus B cube  a+ b= (a + b) (a2 – ab + b2) formula is right or wrong ? for that we need to prove Left hand side (LHS) = Right hand side (RHS). Lets begin with the following steps.

• LHS = a+ b3
• On Solving RHS side we get,
• = (a + b) (a2 – ab + b2)
• On multiplying the a and b separately with (a2 – ab + b2) we get
• = a (a2 – ab + b2) + b(a2 – ab + b2)
• = a3 – a2b + ab2 + a2b – ab+ b3
• = a3 – a2b + a2b + ab2– ab+ b3
• = a3 – 0 + 0 + b3
• = a3 + b3
• Hence proved, LHS = RHS

### List of Some important Algebraic formula

• (a – b)3 = a3 – b3 – 3ab (a – b)
• (a + b)3 = a3 + b3 + 3ab (a + b)
• (a + b)2 = a2 + 2ab + b2
• (a – b)2 = a2 – 2ab + b2
• (a + b) (a – b) = a2 – b2
• (a + b)4 = a4 + 4a3b + 6a2b2 + 4ab3 + b4
• (a – b)4 = a4 – 4a3b + 6a2b2 – 4ab3 + b4
• (x + y + z)2 = x2 + y2 + z2 + 2xy +2yz + 2xz
• (x + y – z)2 = x2 + y2 + z2 + 2xy – 2yz – 2xz
• (x – y + z)2 = x2 + y2 + z2 – 2xy – 2yz + 2xz
• (x – y – z)2 = x2 + y2 + z2 – 2xy + 2yz – 2xz
• x3 + y3 + z3 – 3xyz = (x + y + z) (x2 + y2 + z2 – xy – yz -xz)
• x3 + y3 = (x + y) (x– xy + y2)
• x3 – y3 = (x – y) (x+ xy + y2)

### Solved examples using a3 + b3 = (a + b) (a2 – ab + b2) formula

Now you can understand this a3 + b3 = (a + b) (a2 – ab + b2) formula in detail by solving some good examples

Example 1: Find the value of 1023 + 8by using the A cube plus B cube a3+b3 formula.

Solution: In this question we have to to find: 1023 + 83. Here we will assume that a = 102 and b = 8. So we will replace these A and B with 102 and 8 respectively in the formula of a+ b= (a + b) (a2 – ab + b2).

• 1023+83 = (102+8)(1022 – (102)(8)+82)
= (110) (10404-816+64)
= (110)(9652)
=1061720
• Answer: 1023 + 83 = 1,061,720.

Example 2: Factorize the following expression 8p3 + 27 by using the A cube plus B cube a3+b3 = (a + b) (a2 – ab + b2) formula.

Solution: Here we have to factorize: 8p3 + 27. We will use the a3 + bformula to factorize this expression. Lets assume a = 2p and b = 3. so we can rewrite the given expression in the form of a3 + b3

• 8p3 + 27 = (2p)3 + 33
• a+ b= (a + b) (a2 – ab + b2)
• (2p)3+33 =(2p+3)((2p)2-(2p)(3)+32)
• = (2p+3) (4p2-6p+9)

Example 3: Factorize the following expression : 8p3 + 27q3

Solution : Write (8p3 + 27q3) in the form of (a3 + b3). 8p3 + 27p3  =  (2p)3 + (3q)3

(2p)3 + (3q)3 is in the form of (a+ b3). After comparing (a+ b3) and (2p)+ (3q)3, we get a  =  2p and b  =  3q.

Now we will solve this question this the help of (a+ b3) formula. a3 + b3  =  (a + b)(a2 – ab + b2)

Replace a and b with 2p and 3q respectively.

• (2p)3 + (3q)3  =  (2p + 3q)[(2p)2 – (2p)(3q) + (3q)2]
• 8p3 + 27q3  =  (2p + 3q)(8p2 – 6pq + 9q2)

#### FAQs on A cube plus B cube formula

What Is the Expansion of a+ b3 Formula?

a+ b= (a + b) (a2 – ab + b2)

What Is the a+ b3 Formula in Algebra?

a+ b= (a + b) (a2 – ab + b2)

What is the formula for a cube plus B cube?

a+ b= (a + b) (a2 – ab + b2)

What is the formula of a3 b3?

a+ b= (a + b) (a2 – ab + b2)

What is the formula of A cube plus B cube?

a+ b= (a + b) (a2 – ab + b2)

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