To divide exponents with the same variable base, you subtract the exponent in the denominator from the exponent in the numerator.
Understanding the Rule
When dividing exponential expressions with the same variable base, the rule is simple: subtract the exponents. This is based on the properties of exponents. The YouTube video, "Dividing With Exponents", explains this very clearly, specifically referencing an example of X7 / X5.
Step-by-step Breakdown
- Identify the Common Base: Ensure the variable bases are the same in both the numerator and the denominator.
- Subtract Exponents: Subtract the exponent in the denominator from the exponent in the numerator.
- Example: xm / xn = x(m-n)
Examples
| Division Problem | Solution | Explanation |
|---|---|---|
| x7 / x5 | x2 | 7 - 5 = 2 |
| a10 / a4 | a6 | 10 - 4 = 6 |
| y3 / y | y2 | When a variable has no visible exponent, it's understood to be 1, so 3-1 = 2. |
| b5 / b8 | b-3 or 1/b3 | 5 - 8 = -3. Negative exponents mean the variable is in the denominator. |
Important Considerations
- Negative Exponents: If the exponent resulting from the subtraction is negative, it indicates that the variable is in the denominator, and can be rewritten with a positive exponent by moving it to the denominator.
- Coefficients: When there are coefficients (numbers in front of the variable) in the expression, you divide the coefficients as you normally would and apply the exponent rule separately to the variable. For example (6x5) / (2x2) = 3x3. The coefficient 6 divided by the coefficient 2 is 3, and 5 - 2 = 3, thus the x's result is x3
- Zero Exponent: If the exponent in the numerator and denominator are the same the resulting exponent is 0. Any base to the power of zero is equal to 1 (except for 0 itself).
In Conclusion
Dividing exponents with the same variable base is a matter of subtracting the exponents. Understanding this fundamental property simplifies expressions and helps in more advanced mathematical calculations. This method is consistent and effective across algebra, calculus, and other mathematical disciplines.
