Dividing Powers – Subtracting Exponents with the Same Base

Note: this page covers dividing two powers that share the same base (like x⁵ ÷ x²), a different skill from dividing a plain number by a power of 10, which has its own dedicated page in the Arithmetic Operations section.

Dividing powers with the same base mirrors multiplying them, but in reverse: subtract the exponents. This is the Quotient Law: x^a ÷ x^b = x^a−b. It works because dividing cancels out matching copies of x from the top and bottom, leaving only the leftover copies behind.

The Quotient Law

x^a ÷ x^b = x^a−b, as long as both powers share the same base x.

Simplify 7⁶ ÷ 7² as a single power of 7.

Subtract the exponents: 6 − 2 = 4. So 7⁶ ÷ 7² = 7⁴.

Checking with Real Numbers

Compute 3⁴ ÷ 3² directly, then check it matches 3².

3⁴ ÷ 3² = 81 ÷ 9 = 9. And 3² = 9. They match! 9.

Real-Life Application

Algebra: simplifying fractions where powers of the same variable appear top and bottom.
Scientific notation: dividing two measurements both written with powers of 10.
Computing: comparing storage or bandwidth capacities expressed as powers of 2.

Key Takeaways

To divide powers with the same base, subtract the exponents: x^a ÷ x^b = x^a−b.
This is the reverse of the multiplying-powers rule from the previous page.
When the exponents are equal, this rule produces x⁰ – explored on the next page.

Practice: Dividing Powers

Dividing Powers