a. ta có : \(x^2+y^2=\left(x+y\right)^2-2xy=1^2-2\times\left(-6\right)=13\)

\(x^3+y^3=\left(x+y\right)^3-3xy\left(x+y\right)=1^3-3\times\left(-6\right)\times1=19\)

\(x^5+y^5=\left(x+y\right)\left[x^4-x^3y+x^2y^2-xy^3+y^4\right]\)

\(=\left(x+y\right)\left[\left(x^2+y^2\right)^2-x^2y^2-xy\left(x^2+y^2\right)\right]=1.\left(13^2-\left(-6\right)^2-\left(-6\right).13\right)=211\)

b.\(x^2+y^2=\left(x-y\right)^2+2xy=1+2\times6=13\)

\(x^3-y^3=\left(x-y\right)^3+3xy\left(x-y\right)=1^3+6.3.1=19\)​​

​\(x^5-y^5=\left(x-y\right)\left[\left(x^4+x^3y+x^2y^2+xy^3+y^4\right)\right]\)​​

\(=\left(x-y\right)\left[\left(x^2+y^2\right)^2-x^2y^2+xy\left(x^2+y^2\right)\right]=1.\left(13^2-6^2+6.13\right)=211\)

1/

\(x^2+y^2=\left(x-y\right)^2+2xy=2^2+2.1=6\)

2/

\(x^3-y^3=\left(x-y\right)\left(x^2+y^2+xy\right)=2\left(6+1\right)=14\)

3/

\(x^2-y^2=\left(x-y\right)\left(x+y\right)=2\left(x+y\right)\) (3)

Ta có

\(x^2+y^2=\left(x+y\right)^2-2xy=\left(x+y\right)^2-2=6\)

\(\Rightarrow\left(x+y\right)^2=8\Rightarrow\left(x+y\right)=\pm2\sqrt{2}\) Thay vào (3)

\(\Rightarrow x^2-y^2=2.\pm2\sqrt{2}=\pm4\sqrt{2}\)

4/

\(x^6-y^6=\left(x^3-y^3\right)\left(x^3+y^3\right)\) (4)

Ta có

\(x^3-y^3=14\) (cmt)

Ta có

\(x^3+y^3=\left(x+y\right)\left(x^2+y^2-xy\right)=\left(x+y\right).5=\pm2\sqrt{2}.5=\pm10\sqrt{2}\)

\(\Rightarrow x^6-y^6=\pm10\sqrt{2}.14=\pm140\sqrt{2}\)

9(a-b)^2 - 4(x-y)^2

 có:

x^3+y^3=1 
y^3=1-x^3 
y^6=(1-x^3)^2=x^6-2x^3+1 
=>P=2x^6+3(1-x^3).x^3+x^6-2x^3+1+(1-x^3) 
=2

a)  \(x^2+y^2=\left(x+y\right)^2-2xy=1^2-2.\left(-6\right)=13\)

    \(x^3+y^3=\left(x+y\right)^3-3xy\left(x+y\right)=1^3-3.\left(-6\right).1=19\)

\(x^5+y^5=\left(x^2+y^2\right)\left(x^3+y^3\right)-x^2y^2\left(x+y\right)=13.19-\left(-6\right)^2.1=211\)

b)  \(x^2+y^2=\left(x-y\right)^2+2xy=1^1+2.6=13\)

    \(x^3-y^3=\left(x-y\right)^3+3xy\left(x-y\right)=1^3+3.6.1=19\)

   \(x^5-y^5=\left(x^2+y^2\right)\left(x^3-y^3\right)+x^2y^2\left(x-y\right)=13.19+6^2.1=283\)

2/

2(x⁶+y⁶)-3(x⁴+y⁴)

=2[(x²)³+(y²)³ ] - 3x⁴-3y⁴

=2(x²+y²)(x⁴-x²y²+y⁴)-3x⁴-3y⁴

=2.1(x⁴-x²y²+y⁴)-3x⁴-3y⁴

=2x⁴-2x²y²+2y⁴-3x⁴-3y⁴

=-x⁴-2x²y²-y⁴

=-(x⁴+2x²y²+y⁴)

=-(x²+y²)

=-1

CÓ:     \(x^2+y^2=\left(x+y\right)^2-2xy=3^2-2.2=5\)

CÓ:     \(x^3+y^3=\left(x+y\right)\left(x^2-xy+y^2\right)=3\left(5-2\right)=3.3=9\)

CÓ:     \(x^4+y^4=\left(x^2+y^2\right)^2-2x^2y^2=5^2-2.2^2=25-8=17\)

CÓ:     \(x^5+y^5=\left(x^4+y^4\right)\left(x+y\right)-x^4y-xy^4=3.17-xy\left(x^3+y^3\right)\)

\(=51-2.9=51-18=33\)

CÓ:     \(x^6+y^6=\left(x+y\right)\left(x^5+y^5\right)-xy^5-x^5y\)

\(=3.33-xy\left(x^4+y^4\right)=3.33-2.17\)

\(=99-34=65\)

\(x^2+y^2=\left(x+y\right)^2-2xy=3^2-2.2=9-4=5\)

\(x^3+y^3=\left(x+y\right)^3-3xy\left(x+y\right)=3^3-3.2.3=27-18=9\)

\(x^4+y^4=\left(x+y\right)^4-4xy\left(x^2+y^2\right)-3xy.2xy\)

\(=3^4-4.2.5-3.2.2.2=81-40-24=17\)