الأس الكسري
اختبار: الأس الكسري
1 / 10# الأس الكسري
ما هو الأس الكسري؟
الأس الكسري هو أس يحتوي على:
- بسط. - مقام.
مثل:
<div class="math-display" style="text-align:center;margin:10px 0;direction:ltr;overflow-x:auto;max-width:100%;"><svg aria-label="a^{\frac{1}{2}},\quad a^{\frac{3}{2}},\quad a^{\frac{2}{5}}" data-tex="a^{\frac{1}{2}},\quad a^{\frac{3}{2}},\quad a^{\frac{2}{5}}" style="display:block;margin:0 auto;vertical-align: -0.439ex;" xmlns="http://www.w3.org/2000/svg" width="14.499ex" height="2.757ex" role="img" focusable="false" viewBox="0 -1024.6 6408.7 1218.6" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msup"><g data-mml-node="mi"><use data-c="1D44E" xlink:href="#MJXG-TEX-I-1D44E"></use></g><g data-mml-node="TeXAtom" transform="translate(562,413) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mfrac"><g data-mml-node="mn" transform="translate(220,394) scale(0.707)"><use data-c="31" xlink:href="#MJXG-TEX-N-31"></use></g><g data-mml-node="mn" transform="translate(220,-345) scale(0.707)"><use data-c="32" xlink:href="#MJXG-TEX-N-32"></use></g><rect width="553.6" height="60" x="120" y="220"></rect></g></g></g><g data-mml-node="mo" transform="translate(1173.1,0)"><use data-c="2C" xlink:href="#MJXG-TEX-N-2C"></use></g><g data-mml-node="mstyle" transform="translate(1451.1,0)"><g data-mml-node="mspace"></g></g><g data-mml-node="msup" transform="translate(2617.8,0)"><g data-mml-node="mi"><use data-c="1D44E" xlink:href="#MJXG-TEX-I-1D44E"></use></g><g data-mml-node="TeXAtom" transform="translate(562,413) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mfrac"><g data-mml-node="mn" transform="translate(220,394) scale(0.707)"><use data-c="33" xlink:href="#MJXG-TEX-N-33"></use></g><g data-mml-node="mn" transform="translate(220,-345) scale(0.707)"><use data-c="32" xlink:href="#MJXG-TEX-N-32"></use></g><rect width="553.6" height="60" x="120" y="220"></rect></g></g></g><g data-mml-node="mo" transform="translate(3790.9,0)"><use data-c="2C" xlink:href="#MJXG-TEX-N-2C"></use></g><g data-mml-node="mstyle" transform="translate(4068.9,0)"><g data-mml-node="mspace"></g></g><g data-mml-node="msup" transform="translate(5235.6,0)"><g data-mml-node="mi"><use data-c="1D44E" xlink:href="#MJXG-TEX-I-1D44E"></use></g><g data-mml-node="TeXAtom" transform="translate(562,413) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mfrac"><g data-mml-node="mn" transform="translate(220,394) scale(0.707)"><use data-c="32" xlink:href="#MJXG-TEX-N-32"></use></g><g data-mml-node="mn" transform="translate(220,-345) scale(0.707)"><use data-c="35" xlink:href="#MJXG-TEX-N-35"></use></g><rect width="553.6" height="60" x="120" y="220"></rect></g></g></g></g></g></svg></div>
وللتعامل معه توجد حالتان أساسيتان.
---
# الحالة الأولى: البسط يساوي 1
إذا كان البسط يساوي 1 فإن المقام يمثل **رتبة الجذر**.
القاعدة:
\[ a^{\frac{1}{n}}=\sqrt[n]{a} \]
---
مثال 1
<div class="math-display" style="text-align:center;margin:10px 0;direction:ltr;overflow-x:auto;max-width:100%;"><svg aria-label="4^{\frac12}" data-tex="4^{\frac12}" style="display:block;margin:0 auto;vertical-align: 0;" xmlns="http://www.w3.org/2000/svg" width="2.589ex" height="2.318ex" role="img" focusable="false" viewBox="0 -1024.6 1144.1 1024.6" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msup"><g data-mml-node="mn"><use data-c="34" xlink:href="#MJXG-TEX-N-34"></use></g><g data-mml-node="TeXAtom" transform="translate(533,413) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mfrac"><g data-mml-node="mn" transform="translate(220,394) scale(0.707)"><use data-c="31" xlink:href="#MJXG-TEX-N-31"></use></g><g data-mml-node="mn" transform="translate(220,-345) scale(0.707)"><use data-c="32" xlink:href="#MJXG-TEX-N-32"></use></g><rect width="553.6" height="60" x="120" y="220"></rect></g></g></g></g></g></svg></div>
يساوي:
<div class="math-display" style="text-align:center;margin:10px 0;direction:ltr;overflow-x:auto;max-width:100%;"><svg aria-label="\sqrt{4}" data-tex="\sqrt{4}" style="display:block;margin:0 auto;vertical-align: -0.105ex;" xmlns="http://www.w3.org/2000/svg" width="3.061ex" height="2.398ex" role="img" focusable="false" viewBox="0 -1013.8 1353 1060" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msqrt"><g transform="translate(853,0)"><g data-mml-node="mn"><use data-c="34" xlink:href="#MJXG-TEX-N-34"></use></g></g><g data-mml-node="mo" transform="translate(0,153.8)"><use data-c="221A" xlink:href="#MJXG-TEX-N-221A"></use></g><rect width="500" height="60" x="853" y="893.8"></rect></g></g></g></svg></div>
= 2
> الجذر بدون رقم يعني الجذر التربيعي.
---
مثال 2
<div class="math-display" style="text-align:center;margin:10px 0;direction:ltr;overflow-x:auto;max-width:100%;"><svg aria-label="8^{\frac13}" data-tex="8^{\frac13}" style="display:block;margin:0 auto;vertical-align: -0.05ex;" xmlns="http://www.w3.org/2000/svg" width="2.589ex" height="2.368ex" role="img" focusable="false" viewBox="0 -1024.6 1144.1 1046.6" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msup"><g data-mml-node="mn"><use data-c="38" xlink:href="#MJXG-TEX-N-38"></use></g><g data-mml-node="TeXAtom" transform="translate(533,413) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mfrac"><g data-mml-node="mn" transform="translate(220,394) scale(0.707)"><use data-c="31" xlink:href="#MJXG-TEX-N-31"></use></g><g data-mml-node="mn" transform="translate(220,-345) scale(0.707)"><use data-c="33" xlink:href="#MJXG-TEX-N-33"></use></g><rect width="553.6" height="60" x="120" y="220"></rect></g></g></g></g></g></svg></div>
يساوي:
\[ \sqrt[3]{8} \]
ونسأل:
ما العدد الذي إذا ضرب في نفسه ثلاث مرات أعطى 8؟
الإجابة:
2
لأن:
2 × 2 × 2 = 8
---
مثال 3
<div class="math-display" style="text-align:center;margin:10px 0;direction:ltr;overflow-x:auto;max-width:100%;"><svg aria-label="16^{\frac14}" data-tex="16^{\frac14}" style="display:block;margin:0 auto;vertical-align: -0.05ex;" xmlns="http://www.w3.org/2000/svg" width="3.72ex" height="2.368ex" role="img" focusable="false" viewBox="0 -1024.6 1644.1 1046.6" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msup"><g data-mml-node="mn"><use data-c="31" xlink:href="#MJXG-TEX-N-31"></use><use data-c="36" xlink:href="#MJXG-TEX-N-36" transform="translate(500,0)"></use></g><g data-mml-node="TeXAtom" transform="translate(1033,413) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mfrac"><g data-mml-node="mn" transform="translate(220,394) scale(0.707)"><use data-c="31" xlink:href="#MJXG-TEX-N-31"></use></g><g data-mml-node="mn" transform="translate(220,-345) scale(0.707)"><use data-c="34" xlink:href="#MJXG-TEX-N-34"></use></g><rect width="553.6" height="60" x="120" y="220"></rect></g></g></g></g></g></svg></div>
يساوي:
\[ \sqrt[4]{16} \]
ونسأل:
ما العدد الذي إذا ضرب في نفسه أربع مرات أعطى 16؟
الإجابة:
2
لأن:
2 × 2 × 2 × 2 = 16
---
# الحالة الثانية: البسط لا يساوي 1
إذا كان البسط أكبر من 1:
1. نحسب الجذر أولًا. 2. ثم نرفع الناتج لأس البسط.
القاعدة:
\[ a^{\frac{m}{n}} = \left(\sqrt[n]{a}\right)^m \]
---
مثال 1
<div class="math-display" style="text-align:center;margin:10px 0;direction:ltr;overflow-x:auto;max-width:100%;"><svg aria-label="4^{\frac32}" data-tex="4^{\frac32}" style="display:block;margin:0 auto;vertical-align: 0;" xmlns="http://www.w3.org/2000/svg" width="2.589ex" height="2.317ex" role="img" focusable="false" viewBox="0 -1024.1 1144.1 1024.1" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msup"><g data-mml-node="mn"><use data-c="34" xlink:href="#MJXG-TEX-N-34"></use></g><g data-mml-node="TeXAtom" transform="translate(533,413) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mfrac"><g data-mml-node="mn" transform="translate(220,394) scale(0.707)"><use data-c="33" xlink:href="#MJXG-TEX-N-33"></use></g><g data-mml-node="mn" transform="translate(220,-345) scale(0.707)"><use data-c="32" xlink:href="#MJXG-TEX-N-32"></use></g><rect width="553.6" height="60" x="120" y="220"></rect></g></g></g></g></g></svg></div>
أولًا:
<div class="math-display" style="text-align:center;margin:10px 0;direction:ltr;overflow-x:auto;max-width:100%;"><svg aria-label="\sqrt4=2" data-tex="\sqrt4=2" style="display:block;margin:0 auto;vertical-align: -0.186ex;" xmlns="http://www.w3.org/2000/svg" width="7.209ex" height="2.479ex" role="img" focusable="false" viewBox="0 -1013.8 3186.6 1095.8" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msqrt"><g transform="translate(853,0)"><g data-mml-node="mn"><use data-c="34" xlink:href="#MJXG-TEX-N-34"></use></g></g><g data-mml-node="mo" transform="translate(0,153.8)"><use data-c="221A" xlink:href="#MJXG-TEX-N-221A"></use></g><rect width="500" height="60" x="853" y="893.8"></rect></g><g data-mml-node="mo" transform="translate(1630.8,0)"><use data-c="3D" xlink:href="#MJXG-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(2686.6,0)"><use data-c="32" xlink:href="#MJXG-TEX-N-32"></use></g></g></g></svg></div>
ثم:
<div class="math-display" style="text-align:center;margin:10px 0;direction:ltr;overflow-x:auto;max-width:100%;"><svg aria-label="2^3=8" data-tex="2^3=8" style="display:block;margin:0 auto;vertical-align: -0.186ex;" xmlns="http://www.w3.org/2000/svg" width="6.267ex" height="2.184ex" role="img" focusable="false" viewBox="0 -883.2 2770.1 965.2" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msup"><g data-mml-node="mn"><use data-c="32" xlink:href="#MJXG-TEX-N-32"></use></g><g data-mml-node="mn" transform="translate(533,413) scale(0.707)"><use data-c="33" xlink:href="#MJXG-TEX-N-33"></use></g></g><g data-mml-node="mo" transform="translate(1214.3,0)"><use data-c="3D" xlink:href="#MJXG-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(2270.1,0)"><use data-c="38" xlink:href="#MJXG-TEX-N-38"></use></g></g></g></svg></div>
إذن:
<div class="math-display" style="text-align:center;margin:10px 0;direction:ltr;overflow-x:auto;max-width:100%;"><svg aria-label="4^{\frac32}=8" data-tex="4^{\frac32}=8" style="display:block;margin:0 auto;vertical-align: -0.186ex;" xmlns="http://www.w3.org/2000/svg" width="6.737ex" height="2.502ex" role="img" focusable="false" viewBox="0 -1024.1 2977.7 1106.1" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msup"><g data-mml-node="mn"><use data-c="34" xlink:href="#MJXG-TEX-N-34"></use></g><g data-mml-node="TeXAtom" transform="translate(533,413) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mfrac"><g data-mml-node="mn" transform="translate(220,394) scale(0.707)"><use data-c="33" xlink:href="#MJXG-TEX-N-33"></use></g><g data-mml-node="mn" transform="translate(220,-345) scale(0.707)"><use data-c="32" xlink:href="#MJXG-TEX-N-32"></use></g><rect width="553.6" height="60" x="120" y="220"></rect></g></g></g><g data-mml-node="mo" transform="translate(1421.9,0)"><use data-c="3D" xlink:href="#MJXG-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(2477.7,0)"><use data-c="38" xlink:href="#MJXG-TEX-N-38"></use></g></g></g></svg></div>
---
مثال 2
<div class="math-display" style="text-align:center;margin:10px 0;direction:ltr;overflow-x:auto;max-width:100%;"><svg aria-label="8^{\frac23}" data-tex="8^{\frac23}" style="display:block;margin:0 auto;vertical-align: -0.05ex;" xmlns="http://www.w3.org/2000/svg" width="2.589ex" height="2.368ex" role="img" focusable="false" viewBox="0 -1024.6 1144.1 1046.6" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msup"><g data-mml-node="mn"><use data-c="38" xlink:href="#MJXG-TEX-N-38"></use></g><g data-mml-node="TeXAtom" transform="translate(533,413) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mfrac"><g data-mml-node="mn" transform="translate(220,394) scale(0.707)"><use data-c="32" xlink:href="#MJXG-TEX-N-32"></use></g><g data-mml-node="mn" transform="translate(220,-345) scale(0.707)"><use data-c="33" xlink:href="#MJXG-TEX-N-33"></use></g><rect width="553.6" height="60" x="120" y="220"></rect></g></g></g></g></g></svg></div>
أولًا:
\[ \sqrt[3]{8}=2 \]
ثم:
<div class="math-display" style="text-align:center;margin:10px 0;direction:ltr;overflow-x:auto;max-width:100%;"><svg aria-label="2^2=4" data-tex="2^2=4" style="display:block;margin:0 auto;vertical-align: -0.186ex;" xmlns="http://www.w3.org/2000/svg" width="6.267ex" height="2.185ex" role="img" focusable="false" viewBox="0 -883.9 2770.1 965.9" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msup"><g data-mml-node="mn"><use data-c="32" xlink:href="#MJXG-TEX-N-32"></use></g><g data-mml-node="mn" transform="translate(533,413) scale(0.707)"><use data-c="32" xlink:href="#MJXG-TEX-N-32"></use></g></g><g data-mml-node="mo" transform="translate(1214.3,0)"><use data-c="3D" xlink:href="#MJXG-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(2270.1,0)"><use data-c="34" xlink:href="#MJXG-TEX-N-34"></use></g></g></g></svg></div>
إذن:
<div class="math-display" style="text-align:center;margin:10px 0;direction:ltr;overflow-x:auto;max-width:100%;"><svg aria-label="8^{\frac23}=4" data-tex="8^{\frac23}=4" style="display:block;margin:0 auto;vertical-align: -0.186ex;" xmlns="http://www.w3.org/2000/svg" width="6.737ex" height="2.504ex" role="img" focusable="false" viewBox="0 -1024.6 2977.7 1106.6" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msup"><g data-mml-node="mn"><use data-c="38" xlink:href="#MJXG-TEX-N-38"></use></g><g data-mml-node="TeXAtom" transform="translate(533,413) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mfrac"><g data-mml-node="mn" transform="translate(220,394) scale(0.707)"><use data-c="32" xlink:href="#MJXG-TEX-N-32"></use></g><g data-mml-node="mn" transform="translate(220,-345) scale(0.707)"><use data-c="33" xlink:href="#MJXG-TEX-N-33"></use></g><rect width="553.6" height="60" x="120" y="220"></rect></g></g></g><g data-mml-node="mo" transform="translate(1421.9,0)"><use data-c="3D" xlink:href="#MJXG-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(2477.7,0)"><use data-c="34" xlink:href="#MJXG-TEX-N-34"></use></g></g></g></svg></div>
---
مثال 3
<div class="math-display" style="text-align:center;margin:10px 0;direction:ltr;overflow-x:auto;max-width:100%;"><svg aria-label="16^{\frac54}" data-tex="16^{\frac54}" style="display:block;margin:0 auto;vertical-align: -0.05ex;" xmlns="http://www.w3.org/2000/svg" width="3.72ex" height="2.368ex" role="img" focusable="false" viewBox="0 -1024.6 1644.1 1046.6" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msup"><g data-mml-node="mn"><use data-c="31" xlink:href="#MJXG-TEX-N-31"></use><use data-c="36" xlink:href="#MJXG-TEX-N-36" transform="translate(500,0)"></use></g><g data-mml-node="TeXAtom" transform="translate(1033,413) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mfrac"><g data-mml-node="mn" transform="translate(220,394) scale(0.707)"><use data-c="35" xlink:href="#MJXG-TEX-N-35"></use></g><g data-mml-node="mn" transform="translate(220,-345) scale(0.707)"><use data-c="34" xlink:href="#MJXG-TEX-N-34"></use></g><rect width="553.6" height="60" x="120" y="220"></rect></g></g></g></g></g></svg></div>
أولًا:
\[ \sqrt[4]{16}=2 \]
ثم:
<div class="math-display" style="text-align:center;margin:10px 0;direction:ltr;overflow-x:auto;max-width:100%;"><svg aria-label="2^5=32" data-tex="2^5=32" style="display:block;margin:0 auto;vertical-align: -0.186ex;" xmlns="http://www.w3.org/2000/svg" width="7.398ex" height="2.185ex" role="img" focusable="false" viewBox="0 -883.9 3270.1 965.9" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msup"><g data-mml-node="mn"><use data-c="32" xlink:href="#MJXG-TEX-N-32"></use></g><g data-mml-node="mn" transform="translate(533,413) scale(0.707)"><use data-c="35" xlink:href="#MJXG-TEX-N-35"></use></g></g><g data-mml-node="mo" transform="translate(1214.3,0)"><use data-c="3D" xlink:href="#MJXG-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(2270.1,0)"><use data-c="33" xlink:href="#MJXG-TEX-N-33"></use><use data-c="32" xlink:href="#MJXG-TEX-N-32" transform="translate(500,0)"></use></g></g></g></svg></div>
إذن:
<div class="math-display" style="text-align:center;margin:10px 0;direction:ltr;overflow-x:auto;max-width:100%;"><svg aria-label="16^{\frac54}=32" data-tex="16^{\frac54}=32" style="display:block;margin:0 auto;vertical-align: -0.186ex;" xmlns="http://www.w3.org/2000/svg" width="8.999ex" height="2.504ex" role="img" focusable="false" viewBox="0 -1024.6 3977.7 1106.6" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msup"><g data-mml-node="mn"><use data-c="31" xlink:href="#MJXG-TEX-N-31"></use><use data-c="36" xlink:href="#MJXG-TEX-N-36" transform="translate(500,0)"></use></g><g data-mml-node="TeXAtom" transform="translate(1033,413) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mfrac"><g data-mml-node="mn" transform="translate(220,394) scale(0.707)"><use data-c="35" xlink:href="#MJXG-TEX-N-35"></use></g><g data-mml-node="mn" transform="translate(220,-345) scale(0.707)"><use data-c="34" xlink:href="#MJXG-TEX-N-34"></use></g><rect width="553.6" height="60" x="120" y="220"></rect></g></g></g><g data-mml-node="mo" transform="translate(1921.9,0)"><use data-c="3D" xlink:href="#MJXG-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(2977.7,0)"><use data-c="33" xlink:href="#MJXG-TEX-N-33"></use><use data-c="32" xlink:href="#MJXG-TEX-N-32" transform="translate(500,0)"></use></g></g></g></svg></div>
---
# طريقة أخرى للحساب
يمكن أيضًا كتابة:
<div class="math-display" style="text-align:center;margin:10px 0;direction:ltr;overflow-x:auto;max-width:100%;"><svg aria-label="a^{\frac{m}{n}} = \left(a^{\frac1n}\right)^m" data-tex="a^{\frac{m}{n}} = \left(a^{\frac1n}\right)^m" style="display:block;margin:0 auto;vertical-align: -1.469ex;" xmlns="http://www.w3.org/2000/svg" width="13.16ex" height="4.16ex" role="img" focusable="false" viewBox="0 -1189.1 5816.6 1838.6" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msup"><g data-mml-node="mi"><use data-c="1D44E" xlink:href="#MJXG-TEX-I-1D44E"></use></g><g data-mml-node="TeXAtom" transform="translate(562,413) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mfrac"><g data-mml-node="mi" transform="translate(220,394) scale(0.707)"><use data-c="1D45A" xlink:href="#MJXG-TEX-I-1D45A"></use></g><g data-mml-node="mi" transform="translate(318.3,-345) scale(0.707)"><use data-c="1D45B" xlink:href="#MJXG-TEX-I-1D45B"></use></g><rect width="820.8" height="60" x="120" y="220"></rect></g></g></g><g data-mml-node="mo" transform="translate(1639.9,0)"><use data-c="3D" xlink:href="#MJXG-TEX-N-3D"></use></g><g data-mml-node="msup" transform="translate(2695.7,0)"><g data-mml-node="mrow"><g data-mml-node="mo" transform="translate(0 -0.5)"><use data-c="28" xlink:href="#MJXG-TEX-LO-28"></use></g><g data-mml-node="msup" transform="translate(597,0)"><g data-mml-node="mi"><use data-c="1D44E" xlink:href="#MJXG-TEX-I-1D44E"></use></g><g data-mml-node="TeXAtom" transform="translate(562,413) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mfrac"><g data-mml-node="mn" transform="translate(255.4,394) scale(0.707)"><use data-c="31" xlink:href="#MJXG-TEX-N-31"></use></g><g data-mml-node="mi" transform="translate(220,-345) scale(0.707)"><use data-c="1D45B" xlink:href="#MJXG-TEX-I-1D45B"></use></g><rect width="624.3" height="60" x="120" y="220"></rect></g></g></g><g data-mml-node="mo" transform="translate(1820.1,0) translate(0 -0.5)"><use data-c="29" xlink:href="#MJXG-TEX-LO-29"></use></g></g><g data-mml-node="mi" transform="translate(2450.1,876.6) scale(0.707)"><use data-c="1D45A" xlink:href="#MJXG-TEX-I-1D45A"></use></g></g></g></g></svg></div>
أو
\[ a^{\frac{m}{n}} = \sqrt[n]{a^m} \]
والطريقتان تعطيان نفس النتيجة.
---
# ملخص القواعد
| الأس الكسري | التحويل | |-------------|---------| | <span class="math-inline" dir="ltr" style="display:inline-block;direction:ltr;unicode-bidi:isolate;vertical-align:baseline;white-space:nowrap;margin:0 2px;"><svg aria-label="a^{\frac12}" data-tex="a^{\frac12}" style="display:inline-block;vertical-align: -0.023ex;" xmlns="http://www.w3.org/2000/svg" width="2.654ex" height="2.228ex" role="img" focusable="false" viewBox="0 -974.6 1173.1 984.6" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msup"><g data-mml-node="mi"><use data-c="1D44E" xlink:href="#MJXG-TEX-I-1D44E"></use></g><g data-mml-node="TeXAtom" transform="translate(562,363) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mfrac"><g data-mml-node="mn" transform="translate(220,394) scale(0.707)"><use data-c="31" xlink:href="#MJXG-TEX-N-31"></use></g><g data-mml-node="mn" transform="translate(220,-345) scale(0.707)"><use data-c="32" xlink:href="#MJXG-TEX-N-32"></use></g><rect width="553.6" height="60" x="120" y="220"></rect></g></g></g></g></g></svg></span> | <span class="math-inline" dir="ltr" style="display:inline-block;direction:ltr;unicode-bidi:isolate;vertical-align:baseline;white-space:nowrap;margin:0 2px;"><svg aria-label="\sqrt a" data-tex="\sqrt a" style="display:inline-block;vertical-align: -0.491ex;" xmlns="http://www.w3.org/2000/svg" width="3.127ex" height="2.398ex" role="img" focusable="false" viewBox="0 -843 1382 1060" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msqrt"><g transform="translate(853,0)"><g data-mml-node="mi"><use data-c="1D44E" xlink:href="#MJXG-TEX-I-1D44E"></use></g></g><g data-mml-node="mo" transform="translate(0,-17)"><use data-c="221A" xlink:href="#MJXG-TEX-N-221A"></use></g><rect width="529" height="60" x="853" y="723"></rect></g></g></g></svg></span> | | <span class="math-inline" dir="ltr" style="display:inline-block;direction:ltr;unicode-bidi:isolate;vertical-align:baseline;white-space:nowrap;margin:0 2px;"><svg aria-label="a^{\frac13}" data-tex="a^{\frac13}" style="display:inline-block;vertical-align: -0.023ex;" xmlns="http://www.w3.org/2000/svg" width="2.654ex" height="2.233ex" role="img" focusable="false" viewBox="0 -977.1 1173.1 987.1" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msup"><g data-mml-node="mi"><use data-c="1D44E" xlink:href="#MJXG-TEX-I-1D44E"></use></g><g data-mml-node="TeXAtom" transform="translate(562,365.5) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mfrac"><g data-mml-node="mn" transform="translate(220,394) scale(0.707)"><use data-c="31" xlink:href="#MJXG-TEX-N-31"></use></g><g data-mml-node="mn" transform="translate(220,-345) scale(0.707)"><use data-c="33" xlink:href="#MJXG-TEX-N-33"></use></g><rect width="553.6" height="60" x="120" y="220"></rect></g></g></g></g></g></svg></span> | <span class="math-inline" dir="ltr" style="display:inline-block;direction:ltr;unicode-bidi:isolate;vertical-align:baseline;white-space:nowrap;margin:0 2px;"><svg aria-label="\sqrt[3]{a}" data-tex="\sqrt[3]{a}" style="display:inline-block;vertical-align: -0.491ex;" xmlns="http://www.w3.org/2000/svg" width="3.127ex" height="2.398ex" role="img" focusable="false" viewBox="0 -843 1382 1060" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mroot"><g><g data-mml-node="mi" transform="translate(853,0)"><use data-c="1D44E" xlink:href="#MJXG-TEX-I-1D44E"></use></g></g><g data-mml-node="mn" transform="translate(261.8,344) scale(0.5)"><use data-c="33" xlink:href="#MJXG-TEX-N-33"></use></g><g data-mml-node="mo" transform="translate(0,-17)"><use data-c="221A" xlink:href="#MJXG-TEX-N-221A"></use></g><rect width="529" height="60" x="853" y="723"></rect></g></g></g></svg></span> | | <span class="math-inline" dir="ltr" style="display:inline-block;direction:ltr;unicode-bidi:isolate;vertical-align:baseline;white-space:nowrap;margin:0 2px;"><svg aria-label="a^{\frac14}" data-tex="a^{\frac14}" style="display:inline-block;vertical-align: -0.023ex;" xmlns="http://www.w3.org/2000/svg" width="2.654ex" height="2.228ex" role="img" focusable="false" viewBox="0 -974.6 1173.1 984.6" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msup"><g data-mml-node="mi"><use data-c="1D44E" xlink:href="#MJXG-TEX-I-1D44E"></use></g><g data-mml-node="TeXAtom" transform="translate(562,363) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mfrac"><g data-mml-node="mn" transform="translate(220,394) scale(0.707)"><use data-c="31" xlink:href="#MJXG-TEX-N-31"></use></g><g data-mml-node="mn" transform="translate(220,-345) scale(0.707)"><use data-c="34" xlink:href="#MJXG-TEX-N-34"></use></g><rect width="553.6" height="60" x="120" y="220"></rect></g></g></g></g></g></svg></span> | <span class="math-inline" dir="ltr" style="display:inline-block;direction:ltr;unicode-bidi:isolate;vertical-align:baseline;white-space:nowrap;margin:0 2px;"><svg aria-label="\sqrt[4]{a}" data-tex="\sqrt[4]{a}" style="display:inline-block;vertical-align: -0.491ex;" xmlns="http://www.w3.org/2000/svg" width="3.127ex" height="2.398ex" role="img" focusable="false" viewBox="0 -843 1382 1060" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mroot"><g><g data-mml-node="mi" transform="translate(853,0)"><use data-c="1D44E" xlink:href="#MJXG-TEX-I-1D44E"></use></g></g><g data-mml-node="mn" transform="translate(261.8,333) scale(0.5)"><use data-c="34" xlink:href="#MJXG-TEX-N-34"></use></g><g data-mml-node="mo" transform="translate(0,-17)"><use data-c="221A" xlink:href="#MJXG-TEX-N-221A"></use></g><rect width="529" height="60" x="853" y="723"></rect></g></g></g></svg></span> | | <span class="math-inline" dir="ltr" style="display:inline-block;direction:ltr;unicode-bidi:isolate;vertical-align:baseline;white-space:nowrap;margin:0 2px;"><svg aria-label="a^{\frac{m}{n}}" data-tex="a^{\frac{m}{n}}" style="display:inline-block;vertical-align: -0.023ex;" xmlns="http://www.w3.org/2000/svg" width="3.082ex" height="1.974ex" role="img" focusable="false" viewBox="0 -862.6 1362.1 872.6" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msup"><g data-mml-node="mi"><use data-c="1D44E" xlink:href="#MJXG-TEX-I-1D44E"></use></g><g data-mml-node="TeXAtom" transform="translate(562,363) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mfrac"><g data-mml-node="mi" transform="translate(220,394) scale(0.707)"><use data-c="1D45A" xlink:href="#MJXG-TEX-I-1D45A"></use></g><g data-mml-node="mi" transform="translate(318.3,-345) scale(0.707)"><use data-c="1D45B" xlink:href="#MJXG-TEX-I-1D45B"></use></g><rect width="820.8" height="60" x="120" y="220"></rect></g></g></g></g></g></svg></span> | <span class="math-inline" dir="ltr" style="display:inline-block;direction:ltr;unicode-bidi:isolate;vertical-align:baseline;white-space:nowrap;margin:0 2px;"><svg aria-label="\left(\sqrt[n]{a}\right)^m" data-tex="\left(\sqrt[n]{a}\right)^m" style="display:inline-block;vertical-align: -0.791ex;" xmlns="http://www.w3.org/2000/svg" width="6.791ex" height="2.802ex" role="img" focusable="false" viewBox="0 -889.1 3001.8 1238.6" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msup"><g data-mml-node="mrow"><g data-mml-node="mo" transform="translate(0 -0.5)"><use data-c="28" xlink:href="#MJXG-TEX-SO-28"></use></g><g data-mml-node="mroot" transform="translate(458,0)"><g><g data-mml-node="mi" transform="translate(853,0)"><use data-c="1D44E" xlink:href="#MJXG-TEX-I-1D44E"></use></g></g><g data-mml-node="mi" transform="translate(211.8,338.5) scale(0.5)"><use data-c="1D45B" xlink:href="#MJXG-TEX-I-1D45B"></use></g><g data-mml-node="mo" transform="translate(0,-17)"><use data-c="221A" xlink:href="#MJXG-TEX-N-221A"></use></g><rect width="529" height="60" x="853" y="723"></rect></g><g data-mml-node="mo" transform="translate(1840,0) translate(0 -0.5)"><use data-c="29" xlink:href="#MJXG-TEX-SO-29"></use></g></g><g data-mml-node="mi" transform="translate(2331,576.6) scale(0.707)"><use data-c="1D45A" xlink:href="#MJXG-TEX-I-1D45A"></use></g></g></g></g></svg></span> |
---
# ملاحظات مهمة
- **المقام** يحدد نوع الجذر. - **البسط** يصبح أسًا للناتج بعد استخراج الجذر. - إذا كان البسط يساوي **1**، فالمسألة تتحول مباشرة إلى جذر.
---
ملخص سريع
- <span class="math-inline" dir="ltr" style="display:inline-block;direction:ltr;unicode-bidi:isolate;vertical-align:baseline;white-space:nowrap;margin:0 2px;"><svg aria-label="a^{\frac12}=\sqrt a" data-tex="a^{\frac12}=\sqrt a" style="display:inline-block;vertical-align: -0.491ex;" xmlns="http://www.w3.org/2000/svg" width="8.798ex" height="2.696ex" role="img" focusable="false" viewBox="0 -974.6 3888.7 1191.6" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msup"><g data-mml-node="mi"><use data-c="1D44E" xlink:href="#MJXG-TEX-I-1D44E"></use></g><g data-mml-node="TeXAtom" transform="translate(562,363) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mfrac"><g data-mml-node="mn" transform="translate(220,394) scale(0.707)"><use data-c="31" xlink:href="#MJXG-TEX-N-31"></use></g><g data-mml-node="mn" transform="translate(220,-345) scale(0.707)"><use data-c="32" xlink:href="#MJXG-TEX-N-32"></use></g><rect width="553.6" height="60" x="120" y="220"></rect></g></g></g><g data-mml-node="mo" transform="translate(1450.9,0)"><use data-c="3D" xlink:href="#MJXG-TEX-N-3D"></use></g><g data-mml-node="msqrt" transform="translate(2506.7,0)"><g transform="translate(853,0)"><g data-mml-node="mi"><use data-c="1D44E" xlink:href="#MJXG-TEX-I-1D44E"></use></g></g><g data-mml-node="mo" transform="translate(0,-17)"><use data-c="221A" xlink:href="#MJXG-TEX-N-221A"></use></g><rect width="529" height="60" x="853" y="723"></rect></g></g></g></svg></span> - <span class="math-inline" dir="ltr" style="display:inline-block;direction:ltr;unicode-bidi:isolate;vertical-align:baseline;white-space:nowrap;margin:0 2px;"><svg aria-label="a^{\frac13}=\sqrt[3]{a}" data-tex="a^{\frac13}=\sqrt[3]{a}" style="display:inline-block;vertical-align: -0.491ex;" xmlns="http://www.w3.org/2000/svg" width="8.798ex" height="2.701ex" role="img" focusable="false" viewBox="0 -977.1 3888.7 1194.1" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msup"><g data-mml-node="mi"><use data-c="1D44E" xlink:href="#MJXG-TEX-I-1D44E"></use></g><g data-mml-node="TeXAtom" transform="translate(562,365.5) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mfrac"><g data-mml-node="mn" transform="translate(220,394) scale(0.707)"><use data-c="31" xlink:href="#MJXG-TEX-N-31"></use></g><g data-mml-node="mn" transform="translate(220,-345) scale(0.707)"><use data-c="33" xlink:href="#MJXG-TEX-N-33"></use></g><rect width="553.6" height="60" x="120" y="220"></rect></g></g></g><g data-mml-node="mo" transform="translate(1450.9,0)"><use data-c="3D" xlink:href="#MJXG-TEX-N-3D"></use></g><g data-mml-node="mroot" transform="translate(2506.7,0)"><g><g data-mml-node="mi" transform="translate(853,0)"><use data-c="1D44E" xlink:href="#MJXG-TEX-I-1D44E"></use></g></g><g data-mml-node="mn" transform="translate(261.8,344) scale(0.5)"><use data-c="33" xlink:href="#MJXG-TEX-N-33"></use></g><g data-mml-node="mo" transform="translate(0,-17)"><use data-c="221A" xlink:href="#MJXG-TEX-N-221A"></use></g><rect width="529" height="60" x="853" y="723"></rect></g></g></g></svg></span> - <span class="math-inline" dir="ltr" style="display:inline-block;direction:ltr;unicode-bidi:isolate;vertical-align:baseline;white-space:nowrap;margin:0 2px;"><svg aria-label="a^{\frac14}=\sqrt[4]{a}" data-tex="a^{\frac14}=\sqrt[4]{a}" style="display:inline-block;vertical-align: -0.491ex;" xmlns="http://www.w3.org/2000/svg" width="8.798ex" height="2.696ex" role="img" focusable="false" viewBox="0 -974.6 3888.7 1191.6" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msup"><g data-mml-node="mi"><use data-c="1D44E" xlink:href="#MJXG-TEX-I-1D44E"></use></g><g data-mml-node="TeXAtom" transform="translate(562,363) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mfrac"><g data-mml-node="mn" transform="translate(220,394) scale(0.707)"><use data-c="31" xlink:href="#MJXG-TEX-N-31"></use></g><g data-mml-node="mn" transform="translate(220,-345) scale(0.707)"><use data-c="34" xlink:href="#MJXG-TEX-N-34"></use></g><rect width="553.6" height="60" x="120" y="220"></rect></g></g></g><g data-mml-node="mo" transform="translate(1450.9,0)"><use data-c="3D" xlink:href="#MJXG-TEX-N-3D"></use></g><g data-mml-node="mroot" transform="translate(2506.7,0)"><g><g data-mml-node="mi" transform="translate(853,0)"><use data-c="1D44E" xlink:href="#MJXG-TEX-I-1D44E"></use></g></g><g data-mml-node="mn" transform="translate(261.8,333) scale(0.5)"><use data-c="34" xlink:href="#MJXG-TEX-N-34"></use></g><g data-mml-node="mo" transform="translate(0,-17)"><use data-c="221A" xlink:href="#MJXG-TEX-N-221A"></use></g><rect width="529" height="60" x="853" y="723"></rect></g></g></g></svg></span> - <span class="math-inline" dir="ltr" style="display:inline-block;direction:ltr;unicode-bidi:isolate;vertical-align:baseline;white-space:nowrap;margin:0 2px;"><svg aria-label="a^{\frac{m}{n}}=(\sqrt[n]{a})^m" data-tex="a^{\frac{m}{n}}=(\sqrt[n]{a})^m" style="display:inline-block;vertical-align: -0.566ex;" xmlns="http://www.w3.org/2000/svg" width="12.578ex" height="2.517ex" role="img" focusable="false" viewBox="0 -862.6 5559.5 1112.6" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msup"><g data-mml-node="mi"><use data-c="1D44E" xlink:href="#MJXG-TEX-I-1D44E"></use></g><g data-mml-node="TeXAtom" transform="translate(562,363) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mfrac"><g data-mml-node="mi" transform="translate(220,394) scale(0.707)"><use data-c="1D45A" xlink:href="#MJXG-TEX-I-1D45A"></use></g><g data-mml-node="mi" transform="translate(318.3,-345) scale(0.707)"><use data-c="1D45B" xlink:href="#MJXG-TEX-I-1D45B"></use></g><rect width="820.8" height="60" x="120" y="220"></rect></g></g></g><g data-mml-node="mo" transform="translate(1639.9,0)"><use data-c="3D" xlink:href="#MJXG-TEX-N-3D"></use></g><g data-mml-node="mo" transform="translate(2695.7,0)"><use data-c="28" xlink:href="#MJXG-TEX-N-28"></use></g><g data-mml-node="mroot" transform="translate(3084.7,0)"><g><g data-mml-node="mi" transform="translate(853,0)"><use data-c="1D44E" xlink:href="#MJXG-TEX-I-1D44E"></use></g></g><g data-mml-node="mi" transform="translate(211.8,338.5) scale(0.5)"><use data-c="1D45B" xlink:href="#MJXG-TEX-I-1D45B"></use></g><g data-mml-node="mo" transform="translate(0,-17)"><use data-c="221A" xlink:href="#MJXG-TEX-N-221A"></use></g><rect width="529" height="60" x="853" y="723"></rect></g><g data-mml-node="msup" transform="translate(4466.7,0)"><g data-mml-node="mo"><use data-c="29" xlink:href="#MJXG-TEX-N-29"></use></g><g data-mml-node="mi" transform="translate(422,363) scale(0.707)"><use data-c="1D45A" xlink:href="#MJXG-TEX-I-1D45A"></use></g></g></g></g></svg></span> - احسب **الجذر أولًا** ثم **ارفع الناتج لأس البسط**.
عن هذا الدرس
ما هو الأس الكسري؟ الأس الكسري هو أس يحتوي على: - بسط. - مقام. مثل: وللتعامل معه توجد حالتان أساسيتان. --- الحالة الأولى: البسط يساوي 1 إذا كان البسط… هذا الدرس ضمن مفاهيم القدرات الأساسية لمنهج مفاهيم متنوعة تحضير مفاهيم القدرات في أكاديمية موسى.