التعامل مع الأسس

# التعامل مع الأسس

عندما يكون لدينا نفس الأساس

إذا كان الأساس متساويًا، فهناك أربع قواعد أساسية للتعامل مع الأسس.

---

# 1- الضرب

إذا كان بين الحدين عملية ضرب، نجمع الأسس.

القاعدة:

a^m × a^n = a^(m+n)

### مثال

3² × 3²

= 3^(2+2)

= 3⁴

= 81

---

# 2- القسمة

إذا كان بين الحدين عملية قسمة، نطرح الأسس.

القاعدة:

a^m ÷ a^n = a^(m−n)

### مثال

3² ÷ 3²

= 3^(2−2)

= 3⁰

= 1

---

# 3- الجمع أو الطرح

لا يمكن جمع أو طرح الأسس مباشرة.

بدلاً من ذلك نأخذ العامل المشترك.

---

### مثال (الجمع)

3² + 3²

نأخذ العامل المشترك:

= 3²(1+1)

= 3² × 2

= 9 × 2

= 18

---

### مثال (الطرح)

3² − 3²

= 3²(1−1)

= 3² × 0

= 0

---

# 4- أس مرفوع لأس

إذا كان لدينا أس مرفوع لأس آخر، نضرب الأسين.

القاعدة:

(a^m)^n = a^(m×n)

### مثال

(3²)²

= 3^(2×2)

= 3⁴

= 81

---

# تبسيط الكسور باستخدام الأسس

### مثال

<div class="math-display" style="text-align:center;margin:10px 0;direction:ltr;overflow-x:auto;max-width:100%;"><svg aria-label="\frac{2^{x+1}}{2^x}" data-tex="\frac{2^{x+1}}{2^x}" style="display:block;margin:0 auto;vertical-align: -1.552ex;" xmlns="http://www.w3.org/2000/svg" width="5.274ex" height="4.968ex" role="img" focusable="false" viewBox="0 -1509.9 2331.1 2195.9" 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="mfrac"><g data-mml-node="msup" transform="translate(220,676)"><g data-mml-node="mn"><use data-c="32" xlink:href="#MJXG-TEX-N-32"></use></g><g data-mml-node="TeXAtom" transform="translate(533,363) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJXG-TEX-I-1D465"></use></g><g data-mml-node="mo" transform="translate(572,0)"><use data-c="2B" xlink:href="#MJXG-TEX-N-2B"></use></g><g data-mml-node="mn" transform="translate(1350,0)"><use data-c="31" xlink:href="#MJXG-TEX-N-31"></use></g></g></g><g data-mml-node="msup" transform="translate(671.8,-686)"><g data-mml-node="mn"><use data-c="32" xlink:href="#MJXG-TEX-N-32"></use></g><g data-mml-node="mi" transform="translate(533,289) scale(0.707)"><use data-c="1D465" xlink:href="#MJXG-TEX-I-1D465"></use></g></g><rect width="2091.1" height="60" x="120" y="220"></rect></g></g></g></svg></div>

نطبق قاعدة القسمة:

<div class="math-display" style="text-align:center;margin:10px 0;direction:ltr;overflow-x:auto;max-width:100%;"><svg aria-label="=2^{(x+1)-x}" data-tex="=2^{(x+1)-x}" style="display:block;margin:0 auto;vertical-align: -0.186ex;" xmlns="http://www.w3.org/2000/svg" width="10.072ex" height="2.32ex" role="img" focusable="false" viewBox="0 -943.3 4451.6 1025.3" 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="mo"><use data-c="3D" xlink:href="#MJXG-TEX-N-3D"></use></g><g data-mml-node="msup" transform="translate(1055.8,0)"><g data-mml-node="mn"><use data-c="32" xlink:href="#MJXG-TEX-N-32"></use></g><g data-mml-node="TeXAtom" transform="translate(533,413) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mo"><use data-c="28" xlink:href="#MJXG-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(389,0)"><use data-c="1D465" xlink:href="#MJXG-TEX-I-1D465"></use></g><g data-mml-node="mo" transform="translate(961,0)"><use data-c="2B" xlink:href="#MJXG-TEX-N-2B"></use></g><g data-mml-node="mn" transform="translate(1739,0)"><use data-c="31" xlink:href="#MJXG-TEX-N-31"></use></g><g data-mml-node="mo" transform="translate(2239,0)"><use data-c="29" xlink:href="#MJXG-TEX-N-29"></use></g><g data-mml-node="mo" transform="translate(2628,0)"><use data-c="2212" xlink:href="#MJXG-TEX-N-2212"></use></g><g data-mml-node="mi" transform="translate(3406,0)"><use data-c="1D465" xlink:href="#MJXG-TEX-I-1D465"></use></g></g></g></g></g></svg></div>

<div class="math-display" style="text-align:center;margin:10px 0;direction:ltr;overflow-x:auto;max-width:100%;"><svg aria-label="=2^1" data-tex="=2^1" style="display:block;margin:0 auto;vertical-align: -0.186ex;" xmlns="http://www.w3.org/2000/svg" width="4.508ex" height="2.185ex" role="img" focusable="false" viewBox="0 -883.9 1992.3 965.9" 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="mo"><use data-c="3D" xlink:href="#MJXG-TEX-N-3D"></use></g><g data-mml-node="msup" transform="translate(1055.8,0)"><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="31" xlink:href="#MJXG-TEX-N-31"></use></g></g></g></g></svg></div>

<div class="math-display" style="text-align:center;margin:10px 0;direction:ltr;overflow-x:auto;max-width:100%;"><svg aria-label="=2" data-tex="=2" style="display:block;margin:0 auto;vertical-align: -0.186ex;" xmlns="http://www.w3.org/2000/svg" width="3.52ex" height="1.692ex" role="img" focusable="false" viewBox="0 -666 1555.8 748" 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="mo"><use data-c="3D" xlink:href="#MJXG-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(1055.8,0)"><use data-c="32" xlink:href="#MJXG-TEX-N-32"></use></g></g></g></svg></div>

---

### طريقة أخرى

نفك البسط:

<div class="math-display" style="text-align:center;margin:10px 0;direction:ltr;overflow-x:auto;max-width:100%;"><svg aria-label="2^{x+1}=2^x \times 2" data-tex="2^{x+1}=2^x \times 2" style="display:block;margin:0 auto;vertical-align: -0.186ex;" xmlns="http://www.w3.org/2000/svg" width="13.427ex" height="2.185ex" role="img" focusable="false" viewBox="0 -883.9 5934.6 965.9" 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="TeXAtom" transform="translate(533,413) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJXG-TEX-I-1D465"></use></g><g data-mml-node="mo" transform="translate(572,0)"><use data-c="2B" xlink:href="#MJXG-TEX-N-2B"></use></g><g data-mml-node="mn" transform="translate(1350,0)"><use data-c="31" xlink:href="#MJXG-TEX-N-31"></use></g></g></g><g data-mml-node="mo" transform="translate(2168.9,0)"><use data-c="3D" xlink:href="#MJXG-TEX-N-3D"></use></g><g data-mml-node="msup" transform="translate(3224.7,0)"><g data-mml-node="mn"><use data-c="32" xlink:href="#MJXG-TEX-N-32"></use></g><g data-mml-node="mi" transform="translate(533,413) scale(0.707)"><use data-c="1D465" xlink:href="#MJXG-TEX-I-1D465"></use></g></g><g data-mml-node="mo" transform="translate(4434.4,0)"><use data-c="D7" xlink:href="#MJXG-TEX-N-D7"></use></g><g data-mml-node="mn" transform="translate(5434.6,0)"><use data-c="32" xlink:href="#MJXG-TEX-N-32"></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="\frac{2^x\times2}{2^x}" data-tex="\frac{2^x\times2}{2^x}" style="display:block;margin:0 auto;vertical-align: -1.552ex;" xmlns="http://www.w3.org/2000/svg" width="7.126ex" height="4.61ex" role="img" focusable="false" viewBox="0 -1351.5 3149.9 2037.5" 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="mfrac"><g data-mml-node="mrow" transform="translate(220,676)"><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="mi" transform="translate(533,363) scale(0.707)"><use data-c="1D465" xlink:href="#MJXG-TEX-I-1D465"></use></g></g><g data-mml-node="mo" transform="translate(1209.7,0)"><use data-c="D7" xlink:href="#MJXG-TEX-N-D7"></use></g><g data-mml-node="mn" transform="translate(2209.9,0)"><use data-c="32" xlink:href="#MJXG-TEX-N-32"></use></g></g><g data-mml-node="msup" transform="translate(1081.2,-686)"><g data-mml-node="mn"><use data-c="32" xlink:href="#MJXG-TEX-N-32"></use></g><g data-mml-node="mi" transform="translate(533,289) scale(0.707)"><use data-c="1D465" xlink:href="#MJXG-TEX-I-1D465"></use></g></g><rect width="2909.9" height="60" x="120" y="220"></rect></g></g></g></svg></div>

نختصر:

<div class="math-display" style="text-align:center;margin:10px 0;direction:ltr;overflow-x:auto;max-width:100%;"><svg aria-label="=2" data-tex="=2" style="display:block;margin:0 auto;vertical-align: -0.186ex;" xmlns="http://www.w3.org/2000/svg" width="3.52ex" height="1.692ex" role="img" focusable="false" viewBox="0 -666 1555.8 748" 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="mo"><use data-c="3D" xlink:href="#MJXG-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(1055.8,0)"><use data-c="32" xlink:href="#MJXG-TEX-N-32"></use></g></g></g></svg></div>

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# القواعد الأساسية للأسس

| العملية | القاعدة | |----------|----------| | الضرب | a^m × a^n = a^(m+n) | | القسمة | a^m ÷ a^n = a^(m−n) | | الجمع | أخذ العامل المشترك فقط | | الطرح | أخذ العامل المشترك فقط | | أس لأس | (a^m)^n = a^(mn) |

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# ملاحظات مهمة

- **الضرب ⇒ نجمع الأسس.** - **القسمة ⇒ نطرح الأسس.** - **الجمع أو الطرح ⇒ لا نجمع الأسس، بل نأخذ العامل المشترك.** - **أس مرفوع لأس ⇒ نضرب الأسين.** - جميع هذه القواعد تطبق فقط عندما يكون **الأساس متساويًا**.

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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^m \times a^n = a^{m+n}" data-tex="a^m \times a^n = a^{m+n}" style="display:inline-block;vertical-align: -0.186ex;" xmlns="http://www.w3.org/2000/svg" width="15.91ex" height="1.939ex" role="img" focusable="false" viewBox="0 -775.2 7032.3 857.2" 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="mi"><use data-c="1D44E" xlink:href="#MJXG-TEX-I-1D44E"></use></g><g data-mml-node="mi" transform="translate(562,363) scale(0.707)"><use data-c="1D45A" xlink:href="#MJXG-TEX-I-1D45A"></use></g></g><g data-mml-node="mo" transform="translate(1455.1,0)"><use data-c="D7" xlink:href="#MJXG-TEX-N-D7"></use></g><g data-mml-node="msup" transform="translate(2455.3,0)"><g data-mml-node="mi"><use data-c="1D44E" xlink:href="#MJXG-TEX-I-1D44E"></use></g><g data-mml-node="mi" transform="translate(562,363) scale(0.707)"><use data-c="1D45B" xlink:href="#MJXG-TEX-I-1D45B"></use></g></g><g data-mml-node="mo" transform="translate(3769.3,0)"><use data-c="3D" xlink:href="#MJXG-TEX-N-3D"></use></g><g data-mml-node="msup" transform="translate(4825.1,0)"><g data-mml-node="mi"><use data-c="1D44E" xlink:href="#MJXG-TEX-I-1D44E"></use></g><g data-mml-node="TeXAtom" transform="translate(562,363) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D45A" xlink:href="#MJXG-TEX-I-1D45A"></use></g><g data-mml-node="mo" transform="translate(878,0)"><use data-c="2B" xlink:href="#MJXG-TEX-N-2B"></use></g><g data-mml-node="mi" transform="translate(1656,0)"><use data-c="1D45B" xlink:href="#MJXG-TEX-I-1D45B"></use></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="a^m \div a^n = a^{m-n}" data-tex="a^m \div a^n = a^{m-n}" style="display:inline-block;vertical-align: -0.186ex;" xmlns="http://www.w3.org/2000/svg" width="15.91ex" height="1.939ex" role="img" focusable="false" viewBox="0 -775.2 7032.3 857.2" 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="mi"><use data-c="1D44E" xlink:href="#MJXG-TEX-I-1D44E"></use></g><g data-mml-node="mi" transform="translate(562,363) scale(0.707)"><use data-c="1D45A" xlink:href="#MJXG-TEX-I-1D45A"></use></g></g><g data-mml-node="mo" transform="translate(1455.1,0)"><use data-c="F7" xlink:href="#MJXG-TEX-N-F7"></use></g><g data-mml-node="msup" transform="translate(2455.3,0)"><g data-mml-node="mi"><use data-c="1D44E" xlink:href="#MJXG-TEX-I-1D44E"></use></g><g data-mml-node="mi" transform="translate(562,363) scale(0.707)"><use data-c="1D45B" xlink:href="#MJXG-TEX-I-1D45B"></use></g></g><g data-mml-node="mo" transform="translate(3769.3,0)"><use data-c="3D" xlink:href="#MJXG-TEX-N-3D"></use></g><g data-mml-node="msup" transform="translate(4825.1,0)"><g data-mml-node="mi"><use data-c="1D44E" xlink:href="#MJXG-TEX-I-1D44E"></use></g><g data-mml-node="TeXAtom" transform="translate(562,363) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D45A" xlink:href="#MJXG-TEX-I-1D45A"></use></g><g data-mml-node="mo" transform="translate(878,0)"><use data-c="2212" xlink:href="#MJXG-TEX-N-2212"></use></g><g data-mml-node="mi" transform="translate(1656,0)"><use data-c="1D45B" xlink:href="#MJXG-TEX-I-1D45B"></use></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="(a^m)^n = a^{mn}" data-tex="(a^m)^n = a^{mn}" style="display:inline-block;vertical-align: -0.566ex;" xmlns="http://www.w3.org/2000/svg" width="12.463ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 5508.8 1000" 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="mo"><use data-c="28" xlink:href="#MJXG-TEX-N-28"></use></g><g data-mml-node="msup" transform="translate(389,0)"><g data-mml-node="mi"><use data-c="1D44E" xlink:href="#MJXG-TEX-I-1D44E"></use></g><g data-mml-node="mi" transform="translate(562,363) scale(0.707)"><use data-c="1D45A" xlink:href="#MJXG-TEX-I-1D45A"></use></g></g><g data-mml-node="msup" transform="translate(1621.8,0)"><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="1D45B" xlink:href="#MJXG-TEX-I-1D45B"></use></g></g><g data-mml-node="mo" transform="translate(2795.9,0)"><use data-c="3D" xlink:href="#MJXG-TEX-N-3D"></use></g><g data-mml-node="msup" transform="translate(3851.7,0)"><g data-mml-node="mi"><use data-c="1D44E" xlink:href="#MJXG-TEX-I-1D44E"></use></g><g data-mml-node="TeXAtom" transform="translate(562,363) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D45A" xlink:href="#MJXG-TEX-I-1D45A"></use></g><g data-mml-node="mi" transform="translate(878,0)"><use data-c="1D45B" xlink:href="#MJXG-TEX-I-1D45B"></use></g></g></g></g></g></svg></span> - في الجمع والطرح لا نجمع أو نطرح الأسس، بل نخرج العامل المشترك.