When powers are the same but bases are different, the bases are multiplied first, according to exponent rules.
Exponent Rules Explained
When dealing with exponents, understanding the rules is crucial for accurate calculations. This is especially important when you encounter scenarios where bases are different but the exponents are the same.
Multiplying Bases with the Same Exponent
According to the reference provided, when variable bases are different and the powers (exponents) are the same, the bases are multiplied first. This rule applies not only to variable bases but also to numerical bases.
- Rule: (an) (bn) = (a b)n
Examples
- Numerical Example: 23 33 = (2 3)3 = 63 = 216
- Variable Example: x2 y2 = (x y)2 = (xy)2
Practical Insights
- This rule allows simplifying expressions that otherwise might be difficult to compute.
- The rule applies to any exponent value.
- The order of multiplication doesn't matter for the bases. For example, an bn is the same as bn an.
How This Rule Differs From Other Exponent Rules
- When the bases are the same, we add exponents during multiplication: xm * xn = xm+n.
- When raising a power to a power, we multiply the exponents: (xm)n = xm*n.
- The exponent rules vary based on whether you are multiplying, dividing, or raising powers.
Table Summary
| Scenario | Rule | Example |
|---|---|---|
| Same exponents, different bases | (an) (bn) = (a b)n | 23 * 33 = 63 |
| Same bases, different exponents | xm * xn = xm+n | x2 * x3 = x5 |
| Raising a power to a power | (xm)n = xm*n | (x2)3 = x6 |
In conclusion, the question "When powers are the same, bases are different?" is resolved by understanding that the bases are multiplied first and then that result is raised to the given power.
