Division with exponents

Key Notes :

📘 Division with Exponents

🔹 Rule of Division with Same Bases

When dividing numbers with the same base, subtract the exponents.

👉 Rule:

aᵐ ÷ aⁿ = aᵐ⁻ⁿ

(Where a is the base, m and n are exponents)

🔹 Examples

1️⃣ x⁵ ÷ x² = x⁵⁻² = x³

2️⃣ 7⁴ ÷ 7² = 7⁴⁻² = 7² = 49

3️⃣ a⁶ ÷ a⁶ = a⁶⁻⁶ = a⁰ = 1

⚙️ Bases Must Be the Same

This rule works only when the bases are the same.
If the bases are different, you cannot apply the exponent rule.

🚫 Example (cannot simplify):

3⁴/2⁴ → Bases are different.

💡 Zero and Negative Exponents

If the exponent becomes zero or negative after subtraction, apply these rules:

• a⁰=1
• a⁻ⁿ=1/aⁿ

✅ Example 1: 7³/7³=7³⁻³=7⁰=1

✅ Example 2: 4²/4⁵=4²⁻⁵=4⁻³=1/4³=1/64

🧩 Division of Coefficients and Powers

When dividing numbers with exponents and coefficients,
divide the coefficients and apply the exponent rule to the powers.

✅ Example:

8X⁶/4X²=8/4×X⁶⁻² =2X⁴

📘 Division of Powers with Power

If a power is divided and both have exponents raised again, apply both rules step by step.

✅ Example:

(a³)⁴/(a³)²=a^{(3×4)−(3×2)}=a¹²⁻⁶=a⁶

🔹 Important Notes

• ✨ Bases must be the same to apply the rule.
• ✨ If the base is different, divide normally (do not subtract exponents).
• ✨ Any non-zero number raised to the power of 0 equals 1.
• ✨ When the exponent becomes negative, rewrite it using a reciprocal:
• a⁻ⁿ = 1 / aⁿ

🔹 More Examples

📘 Example 1:

• m⁹ ÷ m⁴ = m⁵

📘 Example 2:

• (2⁶ ÷ 2³) = 2³ = 8

📘 Example 3:

• (a³b²) ÷ (a²b) = a³⁻² b²⁻¹ = ab

🔹 Word Problems

🧩 Example:

• If the area of a square is 5⁶ cm² and the side length is 5³ cm,
• find the missing exponent for the side.

Solution:

• Area ÷ side = 5⁶ ÷ 5³ = 5³
• ✅ Each side = 5³ cm

🔹 Quick Summary Table

Expression	Simplified Form	Rule Used
a⁷ ÷ a²	a⁵	Subtract exponents
4⁵ ÷ 4³	4²	Subtract exponents
a⁵ ÷ a⁵	1	a⁰ = 1
a³ ÷ a⁶	a⁻³ = 1/a³	Negative exponent rule

💡 Tip: Always check the base first! Only subtract exponents when bases are identical.

✨ In short:

• Division with Exponents = Same Base → Subtract Powers

🧠 Memory Trick:

👉 “When you divide powers, just subtract the towers!”

Learn with an example

Let’s practice!🖊️