今日の成果

\sqrt{\dfrac{100^{2}+81^{2}+19^{4}}{2}}=?

10=a 9=bとおく

19=a+b  100^{2}=a^{4}  81^{2}=b^{4}

19^{4}=\left(a+b\right)^{4}

a^{4}+b^{4}=\left(a^{2}+b^{2}\right)^{2}-2a^{2}b^{2}

\left(a^{2}+b^{2}\right)^{2}=\left\{\left(a-b\right)^{2}+2ab\right\}^{2}

\left(a+b\right)^{4}=\left(a^{2}+2ab+b^{2}\right)^{2}

=\left\{\left(a-b\right)^{2}+4ab\right\}^{2}

a-b=10-9=1  ab=Aとおく

100^{2}+81^{2}+19^{4}

=\left\{\left(a-b\right)^{2}+2ab\right\}^{2}-2a^{2}b^{2}

+\left\{\left(a-b\right)^{2}+4ab\right\}^{2}

=\left(1+2A\right)^{2}-2A^{2}+\left(1+4A\right)^{2}

=18A^{2}+12A+2

=2\left(3A+1\right)^{2}

A=10・9=90

従って

与式の？＝２７１