Khử mẫu của biểu thức căn: \(\sqrt{\dfrac{5-2\sqrt{6}}{8}}\).
\(\sqrt{\dfrac{5-2\sqrt{6}}{8}}=\sqrt{\dfrac{2\left(\sqrt{3}-\sqrt{2}\right)^2}{8.2}}=\dfrac{\left|\sqrt{3}-\sqrt{2}\right|\sqrt{2}}{4}=\dfrac{\sqrt{6}-2}{4}.\)
Trục căn thức ở mẫu biểu thức: \(\dfrac{\sqrt{6}-\sqrt{10}}{\sqrt{5}}\).
\(\dfrac{\sqrt{6}-\sqrt{10}}{\sqrt{5}}=\dfrac{\sqrt{5}\left(\sqrt{6}-\sqrt{10}\right)}{\sqrt{5}.\sqrt{5}}=\dfrac{\sqrt{30}-\sqrt{50}}{5}.\)
Tính: \(\frac{2}{\sqrt{13}-\sqrt{11}}+\frac{2}{\sqrt{13}+\sqrt{11}}\)
\(\frac{2}{\sqrt{13}-\sqrt{11}}+\frac{2}{\sqrt{13}+\sqrt{11}}=\frac{2\sqrt{13}+2\sqrt{11}+2\sqrt{13}-2\sqrt{11}}{13-11}\)
\(=\frac{4\sqrt{13}}{2}=2\sqrt{13}\)
Giải phương trình \(\sqrt{2x+3}=1+\sqrt{2}\)
Đáp số: $x = $ @p.sqrt(2)@.
Điều kiện: \(x\ge-\dfrac{3}{2}.\)
\(\sqrt{2x+3}=1+\sqrt{2}\)
\(\Leftrightarrow2x+3=\left(1+\sqrt{2}\right)^2\)
\(\Leftrightarrow2x+3=3+2\sqrt{2}\)
\(\Leftrightarrow2x=2\sqrt{2}\)
\(\Leftrightarrow x=\sqrt{2}\).
p.sqrt = function(s){
return `<span><span class="katex" style="font-size: 1em;"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1.04em; vertical-align: -0.13278em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.90722em;"><span class="svg-align" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="mord" style="padding-left: 0.833em;"><span class="mord"><input style="position:relative; z-index: 100; margin-top: 9px; heihgt: 18px" data-accept="${s}" /></span></span></span><span class="" style="top: -2.86722em;"><span class="pstrut" style="height: 3em;"></span><span class="hide-tail" style="min-width: 0.853em; height: 1.08em;"><svg width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702
c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14
c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54
c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10
s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429
c69,-144,104.5,-217.7,106.5,-221
l0 -0
c5.3,-9.3,12,-14,20,-14
H400000v40H845.2724
s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7
c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z
M834 80h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.13278em;"><span class=""></span></span></span></span></span></span></span></span></span>`;
}
Trục căn thức ở mẫu: \(H=\dfrac{1+2\sqrt{x}}{1-\sqrt{x}}\).
Đáp số: $H=$ @p.c@
p.c = '<span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.24661em;vertical-align:-0.7693300000000001em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.47728em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathit">x</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathit" ><input style="position: relative; font-family: Katex_Math" data-accept="2"/></span><span class="mord mathit">x</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathit" ><input style="position: relative; font-family: Katex_Math" data-accept="3"/></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8002800000000001em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord mathit">x</span></span></span><span style="top:-2.76028em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5, -10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8, -50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0, 35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5, -221c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467 s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422 s-65,47,-65,47z M834 80H400000v40H845z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.23972em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathit" ><input style="position: relative; font-family: Katex_Math" data-accept="1"/></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.7693300000000001em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span>';
Điền số thích hợp vào ô trống:
\(\dfrac{3}{\sqrt{@p.a@}[email protected]@}\) = @p.c@
\(\dfrac{3}{\sqrt{@p.a@}[email protected]@}=\dfrac{3\left(\sqrt{@p.a@}[email protected]@\right)}{@p.a@[email protected]@}=\dfrac{3\sqrt{@p.a@}[email protected]@}{@p.ab2@}\).
p.a = shuffle([5,6,7,11])[0];
p.b = rand(1,1,Math.floor(Math.sqrt(p.a)),[Math.sqrt(p.a-1)]);
params({a: p.a, b:p.b});
p.b2 = p.b*p.b;
p.ab2 = p.a- p.b2;
p.b3 = 3*p.b;
p.c = `<span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.27022em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5842200000000002em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">4</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathit"><input data-accept="${3}"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.90722em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord">${p.a}</span></span></span><span style="top:-2.86722em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5, -10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8, -50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0, 35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5, -221c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467 s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422 s-65,47,-65,47z M834 80H400000v40H845z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.13278em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathit"><input data-accept="${p.b3}"></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span>`;
p.event = function(Zone){ //phím số mobile
Zone.find("input").attr({"inputmode": "numeric"});
};
So sánh:
@p.db[p.t]@ @p.d[p.t]@ || @p.d[1-p.t]@ || = @p.db[1-p.t]@.
Ta thấy \(\sqrt{@p.b2@}-\sqrt{@p.b1@}=\dfrac{\left(\sqrt{@p.b2@}-\sqrt{@p.b1@}\right)\left(\sqrt{@p.b2@}+\sqrt{@p.b1@}\right)}{\sqrt{@p.b2@}+\sqrt{@p.b1@}}=\dfrac{@p.b2@[email protected]@}{\sqrt{@p.b2@}+\sqrt{@p.b1@}}=\dfrac{@p.k@}{\sqrt{@p.b2@}+\sqrt{@p.b1@}}.\)
Tương tự \(\sqrt{@p.a2@}-\sqrt{@p.a1@}=\dfrac{@p.k@}{\sqrt{@p.a2@}+\sqrt{@p.a1@}}.\)
Ta thấy ngay \(\sqrt{@p.b2@}+\sqrt{@p.b1@}>\sqrt{@p.a2@}+\sqrt{@p.a1@}\Rightarrow\dfrac{@p.k@}{\sqrt{@p.b2@}+\sqrt{@p.b1@}}>\dfrac{@p.k@}{\sqrt{@p.a2@}+\sqrt{@p.a1@}}\Rightarrow\sqrt{@p.b2@}-\sqrt{@p.b1@}< \sqrt{@p.a2@}-\sqrt{@p.a1@}.\)
Vậy @p.db[p.t]@ [email protected][p.t]@$ @p.db[1-p.t]@.
p.k = random(2,3);
do {
p.a = random(5,10);
p.b = rand(1,p.a+1,p.a+3,[p.a+2]);
} while (Math.sqrt(p.a)%1 === 0 ||Math.sqrt(p.a+p.k)%1 === 0 || Math.sqrt(p.b)%1 === 0 || Math.sqrt(p.b+p.k)%1 === 0);
p.t = random(0,1);
params({a: p.a, t: p.t, k: p.k, b: p.b});
p.a1 = p.a;
p.a2 = p.a + p.k;
p.b1 = p.b;
p.b2 = p.b + p.k;
p.d = ['<', '>'];
p.db = [
`$\\sqrt{${p.b2}}-\\sqrt{${p.b1}}$`,
`$\\sqrt{${p.a2}}-\\sqrt{${p.a1}}$`
];
Khử mẫu của biểu thức lấy căn: \(\sqrt{\dfrac{21}{5}}\)
Ta có \(\sqrt{\dfrac{21}{5}}=\sqrt{\dfrac{21.5}{5^2}}=\dfrac{\sqrt{105}}{5}\).
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