(1)立方差公式的应用.a^3-b^3=(a-b)(a^2+ab+b^2)
4立方根是4的1/3次方即2的2/3次方,6立方根是6的1/3次方即为2和3的1/3次方根相乘,9立方根是9的1/3次方即3的2/3次方,然后指数运算.
1/1/(4的立方根+6的立方根+9的立方根）
=1/((2^(1/3))^2+2^(1/3)3^(1/3)+(3^(1/3))^2
=[(2^(1/3)-3^(1/3)]/[((2^(1/3))^3-(3^(1/3))^3]
=3^(1/3)-2^(1/3).即为3的立方根减去2的立方根.
（2）平方差公式应用a^2-b^2=(a+b)(a-b)
(1+2√3+√5)/(1+√3)(√3+√5) + (√5+2√7+3)/(√5+√7)(√7+3)
=[(1+2√3+√5)(1-√3)(√3-√5)]/[(1+√3)(√3+√5)(1-√3)(√3-√5)+[(√5+2√7+3)(√5-√7)(√7-3)]/[(√5+√7)(√7+3)(√5-√7)(√7-3)]
=[(1+√3+√3+√5)(1-√3)(√3-√5)]/[(1-3)(3-5)+[(√5+√7+√7+3)(√5-√7)(√7-3)]/(5-7)(7-9)
=[(1-3)(√3-√5)+(1-√3)(3-5)]/4+[(5-7)(√7-3)+(7-9)(√5-√7)]/4
=(-2√3+2√5-2+2√3)/4+(-2√7+6-2√5+2√7)/4
=(2√5-2)/4+(6-2√5)/4
=(2√5-2+6-2√5)/4
=4/4=1
(3)√(2+√3) + √(2-√3) 两边平方
=2+√3+2-√3+2*√(2+√3)*√(2-√3)
=4+2√((2+√3)(2-√3))
=4+2√(2+√3+2-√3)
=4+2*√4=4+2*2=8
再开方,值为2√2
(4)(x+y)^2
=x^2+y^2+2xy
=4-√(10+2√5)+4+√(10+2√5)+2*√{4-√(10+2√5)}√{4+√(10+2√5)}
=8+2*√{[4-√(10+2√5)]*[4+√(10+2√5)
=8+2*√[4^2-√(10+2√5)^2]
=8+2*√(16-10-2√5)
=8+2*√(6-2√5)
=8+2*√(5-2√5+1)
=8+2*√[(√5)^2-2√5+1)]
=8+2*√(√5-1)^2
=8+2*(√5-1)
=6+2√5=(√5)^2+2√5+1=(√5+1)^2
再开方得,x+y=√5+1