我们知道平方运算和开方运算是互逆运算,如:a2±2ab+b2=(a±b)2 , 那么
, 那么如何将双重二次根式![](http://math.21cnjy.com/mml2svg?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmsqrt%3E%3Cmi%3Ea%3C%2Fmi%3E%3Cmo%3E%C2%B1%3C%2Fmo%3E%3Cmn%3E2%3C%2Fmn%3E%3Cmsqrt%3E%3Cmi%3Eb%3C%2Fmi%3E%3C%2Fmsqrt%3E%3C%2Fmsqrt%3E%3C%2Fmath%3E)
![](http://math.21cnjy.com/mml2svg?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmfenced%3E%3Cmrow%3E%3Cmi%3Ea%3C%2Fmi%3E%3Cmo%3E%26gt%3B%3C%2Fmo%3E%3Cmn%3E0%3C%2Fmn%3E%3Cmo%3E%2C%3C%2Fmo%3E%3Cmi%3Eb%3C%2Fmi%3E%3Cmo%3E%26gt%3B%3C%2Fmo%3E%3Cmn%3E0%3C%2Fmn%3E%3Cmo%3E%2C%3C%2Fmo%3E%3Cmi%3Ea%3C%2Fmi%3E%3Cmo%3E%C2%B1%3C%2Fmo%3E%3Cmn%3E2%3C%2Fmn%3E%3Cmsqrt%3E%3Cmi%3Eb%3C%2Fmi%3E%3C%2Fmsqrt%3E%3Cmo%3E%26gt%3B%3C%2Fmo%3E%3Cmn%3E0%3C%2Fmn%3E%3C%2Fmrow%3E%3C%2Fmfenced%3E%3C%2Fmath%3E)
化简呢?如能找到两个数m,n(m>0,n>0),使得
即m+n=a,且使![](http://math.21cnjy.com/mml2svg?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmsqrt%3E%3Cmi%3Em%3C%2Fmi%3E%3C%2Fmsqrt%3E%3C%2Fmath%3E)
![](http://math.21cnjy.com/mml2svg?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmo%3E%C2%B7%3C%2Fmo%3E%3C%2Fmath%3E)
=
即m•n=b,那么
=
∴
=
, 双重二次根式得以化简;
例如化简:
;∵3=1+2且2=1×2,∴
∴![](http://math.21cnjy.com/mml2svg?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmsqrt%3E%3Cmn%3E3%3C%2Fmn%3E%3Cmo%3E%2B%3C%2Fmo%3E%3Cmn%3E2%3C%2Fmn%3E%3Cmsqrt%3E%3Cmn%3E2%3C%2Fmn%3E%3C%2Fmsqrt%3E%3C%2Fmsqrt%3E%3Cmo%3E%3D%3C%2Fmo%3E%3Cmn%3E1%3C%2Fmn%3E%3Cmo%3E%2B%3C%2Fmo%3E%3Cmsqrt%3E%3Cmn%3E2%3C%2Fmn%3E%3C%2Fmsqrt%3E%3C%2Fmath%3E)
由此对于任意一个二次根式只要可以将其化成
的形式,且能找到m,n(m>0,n>0)使得m+n=a,且m•n=b,那么这个双重二次根式一定可以化简为一个二次根式.请同学们通过阅读上述材料,完成下列问题: