化簡：(8-根號63)的平方根

解題

1.化簡：\sqrt{8-\sqrt{63}}
解：\sqrt{8-\sqrt{63}}
=\sqrt{\frac{1}{2}}×\sqrt{16-2\sqrt{63}}
=\frac{1}{2}\sqrt{2}×\sqrt{\sqrt{9}^2+\sqrt{7}^2-2×\sqrt{9}×\sqrt{7}}
=\frac{1}{2}\sqrt{2}×\sqrt{(\sqrt{9}-\sqrt{7})^2}
=\frac{1}{2}\sqrt{2}×(\sqrt{9}-\sqrt{7})
=\frac{1}{2}×\sqrt{2}×(3-\sqrt{7})
=\frac{3}{2}\sqrt{2}-\frac{1}{2}\sqrt{14}

2.化簡：\sqrt{6-\sqrt{35}}
解：\sqrt{6-\sqrt{35}}
=\sqrt{\frac{1}{2}}×\sqrt{12-2\sqrt{35}}
=\frac{1}{2}\sqrt{2}×\sqrt{\sqrt{7}^2+\sqrt{5}^2-2×\sqrt{7}×\sqrt{5}}
=\frac{1}{2}\sqrt{2}×\sqrt{(\sqrt{7}-\sqrt{5})^2}
=\frac{1}{2}\sqrt{2}×(\sqrt{7}-\sqrt{5})
=\frac{1}{2}\sqrt{14}-\frac{1}{2}\sqrt{10}

3.化簡：\sqrt{12-\sqrt{143}}
解：\sqrt{12-\sqrt{143}}
=\sqrt{\frac{1}{2}}×\sqrt{24-2\sqrt{143}}
=\frac{1}{2}\sqrt{2}×\sqrt{\sqrt{13}^2+\sqrt{11}^2-2×\sqrt{13}×\sqrt{11}}
=\frac{1}{2}\sqrt{2}×\sqrt{(\sqrt{13}-\sqrt{11})^2}
=\frac{1}{2}\sqrt{2}×(\sqrt{13}-\sqrt{11})
=\frac{1}{2}\sqrt{26}-\frac{1}{2}\sqrt{22}

探究

\sqrt{8-\sqrt{63}}=\sqrt{\sqrt{64}-\sqrt{63}}
=\sqrt{8-\sqrt{9}×\sqrt{7}}
=\color{#0000ff}{\sqrt{8-\sqrt{8+1}×\sqrt{8-1}}}

\sqrt{6-\sqrt{35}}=\sqrt{\sqrt{36}-\sqrt{63}}
=\color{#0000ff}{\sqrt{6-\sqrt{6+1}×\sqrt{6-1}}}

\sqrt{12-\sqrt{143}}=\sqrt{\sqrt{144}-\sqrt{63}}
=\color{#0000ff}{\sqrt{12-\sqrt{12+1}×\sqrt{12-1}}}

一般形式：\color{#0000ff}{\sqrt{a-\sqrt{a+1}×\sqrt{a-1}}}

\color{#ff0000}{化簡過程}
\sqrt{a-\sqrt{a+1}×\sqrt{a-1}}
\color{#009999}{=\sqrt{\frac{1}{2}}×\sqrt{2a-2\sqrt{a+1}×\sqrt{a-1}}}
\color{#009999}{=\frac{1}{2}\sqrt{2}×\sqrt{(a+1)+(a-1)-2\sqrt{a+1}×\sqrt{a-1}}}
\color{#009999}{=\frac{1}{2}\sqrt{2}×\sqrt{\sqrt{a+1}^2+\sqrt{a-1}^2-2\sqrt{a+1}×\sqrt{a-1}}}
\color{#009999}{=\frac{1}{2}\sqrt{2}×\sqrt{(\sqrt{a+1}-\sqrt{a-1})^2}}
因為a+1>a-1，所以，
原式\color{#009999}{=\frac{1}{2}\sqrt{2}×(\sqrt{a+1}-\sqrt{a-1})}
\color{#ff0000}{再根據(jù)具體的數(shù)，進一步化簡}

練習(xí)

\sqrt{11-\sqrt{120}}
\sqrt{9-\sqrt{80}}
\sqrt{7-\sqrt{48}}
\color{#ff0000}{當(dāng)一般形式里，a為奇數(shù)時，是不是更容易化簡}

再探究

\color{#0000ff}{\sqrt{a-\sqrt{a+1}×\sqrt{a-1}}}\color{#ff0000}{（a為奇數(shù)）}
\color{#ff0000}{化簡過程}
{設(shè)a=2k+1}
原式\color{#009999}{=\sqrt{2k+1-\sqrt{2k+1+1}×\sqrt{2k+1-1}}}
\color{#009999}{=\sqrt{2k+1-\sqrt{2(k+1)}×\sqrt{2k}}}
\color{#009999}{=\sqrt{2k+1-\sqrt{2}\times\sqrt{k+1}\times\sqrt{2}\times\sqrt{k}}}
\color{#009999}{=\sqrt{k+k+1-2\times\sqrt{k+1}\times\sqrt{k}}}
\color{#009999}{=\sqrt{\sqrt{k}^2+\sqrt{k+1}^2-2\times\sqrt{k+1}\times\sqrt{k}}}
\color{#009999}{=\sqrt{(\sqrt{k}-\sqrt{k+1})^2}}
\color{#009999}{=\sqrt{k+1}-\sqrt{k}}

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