\(a,\left(a+b\right)^3-\left(a-b\right)^3-6a^2b\)

\(=a^3+3a^2b+3ab^2+b^3-a^3+3a^2b-3ab^2+b^3-6a^2b\)

\(=2b^3\)

\(b,\left(a+b\right)^3+\left(a-b\right)^3-6ab^2\)

\(=a^3+3a^2b+3ab^2+b^3+a^3-3a^2b+3ab^2-b^3-6ab^2\)

\(=2a^3\)

ai lam nhanh nhat thi minh k cho

a) (a+b)³+(a-b)³=a³+3a²b+3ab²+b³+a³-3a²b+3ab²-b³

                          =2a³+6ab²

b) (a + b + c)² + (a − b − c)² + (b − c − a)² + (c − a − b)²

=a²+b²+c²+2ab+2bc+2ca+a²+b²+c²-2ab+2bc-2ac+a²+b²+c²-2bc+2ca-2ba+a²+b²+c²-2ca+2ab-2cb

=4a²+4b²+4c²

a) Ta có: \(\left(a+b\right)^3+\left(a-b\right)^3\)

\(=\left(a+b+a-b\right)\left[\left(a+b\right)^2-\left(a+b\right)\left(a-b\right)+\left(a-b\right)^2\right]\)

\(=2a\cdot\left(a^2+2ab+b^2-a^2+b^2+a^2-2ab+b^2\right)\)

\(=2a\cdot\left(a^2+3b^2\right)\)

\(=2a^3+6ab^2\)

\(a,=a^3+3a^2b+3ab^2+b^3-a^3+3a^2b-3ab^2+b^3-2b^3=6a^2b\\ b,=\left(6x+1-6x+1\right)^2=2^2=4\)

Rút gọn biểu thức: (a+b)\(^3\) – (a–b)\(^3\)

Ta có: \(\left(a+b\right)^3-\left(a-b\right)^3\)

= \(a^3+3a^2b+3ab^2+b^3-a^3-3a^2b+3ab^2-b^3\)

= \(6ab^2\)

(a+b)³-(a-b)³ = a³+3a²b+3ab²+b³-a³+3a²b-3ab²+b³

=6a²b+2b³ =2b(3a² +b²)

\(a,\left(a+b\right)^2-\left(a-b\right)^2\)

\(=a^2+2ab+b^2-a^2+2ab-b^2\)

\(=4ab\)

\(b,\left(a+b\right)^3-\left(a-b\right)-\left(2b\right)^3\)

\(=a^3+3a^2b+3ab^2+b^3-a+b-8b^3\)

a) \(\left(a+b\right)^2-\left(a-b\right)^2\)

      \(\left(a+b-a+b\right)\left(a+b+a-b\right)\)

     \(\left(2b\right)\left(2a\right)\)

     \(4ab\)

b) \(\left(a+b\right)^3-\left(a-b\right)-\left(2b\right)^3\)

    \(a^3+3a^2b+3ab^2+b^3-a+b-8b^3\)

    \(a\left(a^2-1\right)+3\left(a^2b+ab^2\right)+b\left(b^2+1-8b^2\right)\)

   \(a\left(a-1\right)\left(a+1\right)+3\left[ab\left(a+b\right)\right]+b\left(-7b^2+1\right)\)

Câu hỏi của Adolf Hitler - Toán lớp 8 - Học toán với OnlineMath

Áp dụng hằng đẳng thức dưới dạng 

\(x^3+y^3=\left(x+y\right)^3-3xy\left(x+y\right)\)

\(\left(a+b+c\right)^3+\left(a-b-c\right)^3=\left(2a\right)^3-3\left(a+b+c\right)\left(a-b-c\right).2a\)

\(\left(b-c-a\right)^3+\left(c-a-b\right)^3=\left(-2a\right)^3-3\left(b-c-a\right)\left(c-a-b\right).\left(-2a\right)\)

\(\Rightarrow\left(a+b+c\right)^3+\left(a-b-c\right)^3+\left(b-c-a\right)^3+\left(c-a-b\right)^3\)

\(=\left(2\right)^3+\left(-2a\right)^3-6a\left[a+\left(b+c\right)\right]\left[a-\left(b+c\right)\right]+6a\left[-a+\left(b-c\right)\right]\left[-a-\left(b-c\right)\right]\)

\(=-6a\left\{a^2-\left(b+c\right)^2-\left[\left(-a\right)^2-\left(b-c\right)^2\right]\right\}\)

\(=-6a\left\{a^2-a^2+\left(b-c\right)^2-\left(b+c\right)^2\right\}\)

\(=-6a\left[b-c+b+c\right]\left[b-c-\left(b+c\right)\right]=-6a.2b.\left(-2c\right)\)

\(=24abc\)

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