和四次方

主验证

和四次方可直接计算验证：

${\displaystyle (a+b)^{4}\,\!}$ 

${\displaystyle =(a+b)(a+b)(a+b)(a+b)\,\!}$ 

${\displaystyle =a(a+b)(a+b)(a+b)+b(a+b)(a+b)(a+b)\,\!}$ 

${\displaystyle =(a^{2}+ab)(a+b)(a+b)+(ab+b^{2})(a+b)(a+b)\,\!}$ 

${\displaystyle =a(a^{2}+ab)(a+b)+b(a^{2}+ab)(a+b)+a(ab+b^{2})(a+b)+b(ab+b^{2})(a+b)\,\!}$ 

${\displaystyle =a^{3}(a+b)+a^{2}b(a+b)+a^{2}b(a+b)+ab^{2}(a+b)+a^{2}b(a+b)+ab^{2}(a+b)+ab^{2}(a+b)+b^{3}(a+b)\,\!}$ 

${\displaystyle =a^{4}+a^{3}b+a^{3}b+a^{2}b^{2}+a^{3}b+a^{2}b^{2}+a^{2}b^{2}+ab^{3}+a^{3}b+a^{2}b^{2}+a^{2}b^{2}+ab^{3}+a^{2}b^{2}+ab^{3}+ab^{3}+b^{4}\,\!}$ 

${\displaystyle =a^{4}+4a^{3}b+6a^{2}b^{2}+4ab^{3}+b^{4}\,\!}$ 

运用和立方

和四次方亦可运用和立方验证，首先要知道和立方的公式是：

${\displaystyle (a+b)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3}\,\!}$ 

然后，利用和立方计算出和四次方：

${\displaystyle (a+b)^{4}\,\!}$ 

${\displaystyle =(a+b)^{3}(a+b)\,\!}$ 

${\displaystyle =(a^{3}+3a^{2}b+3ab^{2}+b^{3})(a+b)\,\!}$ 

${\displaystyle =a(a^{3}+3a^{2}b+3ab^{2}+b^{3})+b(a^{3}+3a^{2}b+3ab^{2}+b^{3})\,\!}$ 

${\displaystyle =a^{4}+3a^{3}b+3a^{2}b^{2}+ab^{3}+a^{3}b+3a^{2}b^{2}+3ab^{3}+b^{4}\,\!}$ 

${\displaystyle =a^{4}+4a^{3}b+6a^{2}b^{2}+4ab^{3}+b^{4}\,\!}$ 

运用三数和平方

和四次方亦可运用三数和平方验证，首先要知道和平方的公式是：

${\displaystyle (a+b)^{2}=a^{2}+2ab+b^{2}\,\!}$ 

• 下一步要知道三数和平方的公式是：

${\displaystyle (a+b+c)^{2}=a^{2}+b^{2}+c^{2}+2ab+2bc+2ca\,\!}$ 

然后利用三数和平方的恒等式，计算和平方的平方计算出和四次方：

${\displaystyle (a+b)^{4}\,\!}$ 

${\displaystyle =[(a+b)^{2}]^{2}\,\!}$ 

${\displaystyle =(a^{2}+2ab+b^{2})^{2}\,\!}$ 

${\displaystyle =(a^{2})^{2}+(2ab)^{2}+(b^{2})^{2}+2(a^{2})(2ab)+2(2ab)(b^{2})+2(b^{2})(a^{2})\,\!}$ 

${\displaystyle =a^{4}+4a^{2}b^{2}+b^{4}+4a^{3}b+4ab^{3}+2a^{2}b^{2}\,\!}$ 

${\displaystyle =a^{4}+4a^{3}b+6a^{2}b^{2}+4ab^{3}+b^{4}\,\!}$