化简：(8-根号63)的平方根

解题

1.化简： $\sqrt{8-\sqrt{63}}$
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.化简： $\sqrt{6-\sqrt{35}}$
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.化简： $\sqrt{12-\sqrt{143}}$
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： $\color{#0000ff}{\sqrt{a-\sqrt{a+1}×\sqrt{a-1}}}$

$\color{#ff0000}{化简过程}$
$\sqrt{a-\sqrt{a+1}×\sqrt{a-1}}$
$\color{#009999}{=\sqrt{\frac{1}{2}}×\sqrt{2a-2\sqrt{a+1}×\sqrt{a-1}}}$
$\color{#009999}{=\frac{1}{2}\sqrt{2}×\sqrt{(a+1)+(a-1)-2\sqrt{a+1}×\sqrt{a-1}}}$
$\color{#009999}{=\frac{1}{2}\sqrt{2}×\sqrt{\sqrt{a+1}^2+\sqrt{a-1}^2-2\sqrt{a+1}×\sqrt{a-1}}}$
$\color{#009999}{=\frac{1}{2}\sqrt{2}×\sqrt{(\sqrt{a+1}-\sqrt{a-1})^2}}$
因为a+1>a-1，所以，
原式 $\color{#009999}{=\frac{1}{2}\sqrt{2}×(\sqrt{a+1}-\sqrt{a-1})}$
$\color{#ff0000}{再根据具体的数，进一步化简}$

练习

$\sqrt{11-\sqrt{120}}$
$\sqrt{9-\sqrt{80}}$
$\sqrt{7-\sqrt{48}}$
$\color{#ff0000}{当一般形式里，a为奇数时，是不是更容易化简}$

再探究

$\color{#0000ff}{\sqrt{a-\sqrt{a+1}×\sqrt{a-1}}}\color{#ff0000}{（a为奇数）}$
$\color{#ff0000}{化简过程}$
${设a=2k+1}$
原式 $\color{#009999}{=\sqrt{2k+1-\sqrt{2k+1+1}×\sqrt{2k+1-1}}}$
$\color{#009999}{=\sqrt{2k+1-\sqrt{2(k+1)}×\sqrt{2k}}}$
$\color{#009999}{=\sqrt{2k+1-\sqrt{2}\times\sqrt{k+1}\times\sqrt{2}\times\sqrt{k}}}$
$\color{#009999}{=\sqrt{k+k+1-2\times\sqrt{k+1}\times\sqrt{k}}}$
$\color{#009999}{=\sqrt{\sqrt{k}^2+\sqrt{k+1}^2-2\times\sqrt{k+1}\times\sqrt{k}}}$
$\color{#009999}{=\sqrt{(\sqrt{k}-\sqrt{k+1})^2}}$
$\color{#009999}{=\sqrt{k+1}-\sqrt{k}}$