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Bài 3:
Ta có:
\(81^8-1=\left(9^2\right)^8-1=\left[\left(3^2\right)^2\right]^8-1=3^{32}-1\)
\(=\left(3^4-1\right)\left(3^4+1\right)\left(3^8+1\right)\left(3^{16}+1\right)\)
Do đó:
\(A=3^4-1=80\)
\(A=\frac{1}{2}x^4+x^2y^2+\frac{1}{2}y^4-2x^2y^2\)
\(=\frac{1}{2}\left(x^4-2x^2y^2+y^4\right)=\frac{1}{2}\left(x^2-y^2\right)^2=\frac{1}{2}.4^2=8\)
Câu 1: Ta có: A = \(x^3+y^3+3xy=x^3+y^3+3xy\times1=x^3+y^3+3xy\left(x+y\right)\)
\(=\left(x+y\right)^3=1^3=1\)
Câu 2: Ta có: \(B=x^3-y^3-3xy=\left(x-y\right)\left(x^2+xy+y^2\right)-3xy\)
\(=x^2+xy+y^2-3xy=x^2-2xy+y^2=\left(x-y\right)^2=1^2=1\)
Câu 3: Ta có: \(C=x^3+y^3+3xy\left(x^2+y^2\right)-6x^2.y^2\left(x+y\right)\)
\(=x^3+y^3+3xy\left(x^2+2xy+y^2-2xy\right)+6x^2y^2\)
\(=x^3+y^3+3xy\left(x+y\right)^2-3xy.2xy+6x^2y^2\)
\(=x^3+y^3+3xy.1-6x^2y^2+6x^2y^3\)
\(=x^3+y^3+3xy\left(x+y\right)=\left(x+y\right)^3=1^3=1\)
\(A=3\left(x^2+y^2\right)-2\left(x^3+y^3\right)\)
\(=3x^2+3y^2-2\left(x+y\right)\left(x^2-xy+y^2\right)\)
\(=3x^2+3y^2-2.1\left(x^2-xy+y^2\right)\)
\(=3x^2+3y^2-2x^2+2xy-2y^2\)
\(=x^2+2xy+y^2=\left(x+y\right)^2=1^2=1\)
\(B=x^3+y^3+3xy\left(x^2+y^2\right)+6x^2y^2\left(x+y\right)\)
\(=x^3+y^3+3xy\left[\left(x+y\right)^2-2xy\right]+6x^2y^2.1\)
\(=x^3+y^3+3xy\left(x+y\right)^2-6x^2y^2+6x^2y^2\)
\(=\left(x+y\right)\left(x^2-xy+y^2\right)+3xy\)
\(=x^2-xy+y^2+3xy\)
\(=x^2+2xy+y^2=\left(x+y\right)^2=1^2=1\)
a) Ta có:
\(x+y=1\)
\(\Rightarrow\left(x+y\right)^3=1\)
\(\Rightarrow x^3+y^3+3xy\left(x+y\right)=1\)
\(\Rightarrow x^3+y^3+3xy=1\)
\(\Rightarrow P=1\)
b) \(Q=x^3+y^3+3xy\left(x^2+y^2\right)+6x^2y^2\left(x+y\right)\)
\(Q=\left(x+y\right)\left(x^2-xy+y^2\right)+3xy\left[\left(x+y\right)^2-2xy\right]+6x^2y^2\left(x+y\right)\)
\(Q=\left(x+y\right)\left[\left(x+y\right)^2-3xy\right]+3xy\left[\left(x+y\right)^2-2xy\right]+6x^2y^2\left(x+y\right)\)
Thay x + y = 1 vào Q
\(Q=1-3xy+3xy\left(1-2xy\right)+6x^2y^2\)
\(Q=1-3xy+3xy-6x^2y^2+6x^2y^2\)
\(Q=1\)
Sửa đề: x+y=1
\(A=\left(x+y\right)^3-3xy\left(x+y\right)+3xy\left[\left(x+y\right)^2-2xy\right]+6x^2y^2\)
\(=1-3xy+3xy\left[1-2xy\right]+6x^2y^2\)
=1