The Quotient of Powers Property in algebra provides a rule for simplifying expressions where you are dividing powers that have the same base.
Understanding the Quotient of Powers Property
The Quotient of Powers Property states that when dividing powers with the same base, the exponents are subtracted, assuming the base is not equal to 0. Mathematically, this can be represented as:
na ÷ nb = n(a − b)
Where:
- n is the base (and n ≠ 0)
- a and b are exponents
Explanation
In simpler terms, if you have the same base raised to different powers and you're dividing them, you can simplify the expression by keeping the base and subtracting the exponent in the denominator from the exponent in the numerator.
Examples
Here are some examples to illustrate the Quotient of Powers Property:
- Example 1: x5 / x2 = x(5-2) = x3
- Example 2: 27 / 24 = 2(7-4) = 23 = 8
- Example 3: y3 / y9 = y(3-9) = y-6 = 1/y6
Why does this work?
Consider x5 / x2. This can be written as (x x x x x) / (x x). Two of the x's in the numerator cancel out with the two x's in the denominator, leaving x x * x, which is x3. This demonstrates the subtraction of the exponents.
Practical Insights
- Simplification: The Quotient of Powers Property simplifies complex algebraic expressions, making them easier to understand and work with.
- Solving Equations: It aids in solving equations involving exponents.
- Applications: This property is used extensively in scientific notation, engineering, and computer science.
Table Summarizing the Property
| Property | Description | Formula | Example |
|---|---|---|---|
| Quotient of Powers | Dividing powers with the same base | na ÷ nb = n(a − b) | x7 / x3 = x4 |
