Tell me the formula of (a+b) whole cube. I can’t remember it…
The Correct Answer and Explanation is:
Formula:
(a + b)³ = a³ + 3a²b + 3ab² + b³
Explanation:
The expansion of the expression (a+b)3(a + b)^3(a+b)3 follows the algebraic identity used to multiply a binomial three times. This means multiplying (a+b)(a + b)(a+b) by itself three times:
(a+b)(a+b)(a+b)(a + b)(a + b)(a + b)(a+b)(a+b)(a+b)
To understand the structure, begin by first multiplying the first two binomials:
(a+b)(a+b)=a2+2ab+b2(a + b)(a + b) = a^2 + 2ab + b^2(a+b)(a+b)=a2+2ab+b2
Then, multiply the result by the third (a+b)(a + b)(a+b) term:
(a2+2ab+b2)(a+b)(a^2 + 2ab + b^2)(a + b)(a2+2ab+b2)(a+b)
Now distribute each term in the first trinomial over the second binomial:
- a2⋅a=a3a^2 \cdot a = a^3a2⋅a=a3
- a2⋅b=a2ba^2 \cdot b = a^2ba2⋅b=a2b
- 2ab⋅a=2a2b2ab \cdot a = 2a^2b2ab⋅a=2a2b
- 2ab⋅b=2ab22ab \cdot b = 2ab^22ab⋅b=2ab2
- b2⋅a=ab2b^2 \cdot a = ab^2b2⋅a=ab2
- b2⋅b=b3b^2 \cdot b = b^3b2⋅b=b3
Now combine like terms:
- a3a^3a3
- a2b+2a2b=3a2ba^2b + 2a^2b = 3a^2ba2b+2a2b=3a2b
- ab2+2ab2=3ab2ab^2 + 2ab^2 = 3ab^2ab2+2ab2=3ab2
- b3b^3b3
Final result:
(a+b)3=a3+3a2b+3ab2+b3(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3(a+b)3=a3+3a2b+3ab2+b3
This expansion illustrates the symmetry in binomial cube identities. Each term reflects a combination of the powers of aaa and bbb that add up to 3. The coefficients (1, 3, 3, 1) match the third row of Pascal’s Triangle, which provides a pattern for binomial expansions. This identity is foundational in algebra and simplifies solving problems involving cubic expressions in both pure mathematics and applied contexts such as physics and engineering.
