How Do You Solve a Problem When You Have Different Bases With the Same Exponents?
When multiplying or dividing different bases with the same exponent, combine the bases, and keep the exponent the same. For example, X raised to the third power times Y raised to the third power becomes the product of X times Y raised to the third power. When adding or subtracting different bases with the same power, evaluate the exponents first, and then perform the summation.
The reason it is possible to combine the bases when dealing with multiplication and division is due to the commutative property of multiplication and division, which means that changing the order of the operations doesn't change the result. For instance, X raised to the third power times Y raised to the third power is the same as XXXYYY, which is the same as XY raised to the third power. However, if the exponents are different, this doesn't hold true.
In multiplication and division, when the bases are the same and the exponents are different, the exponents can be added or subtracted, respectively. For example, X raised to the third power times X raised to the second power is the same as X raised to the fifth power. In this example, the exponents are added because it is a multiplication problem. For division problems, the exponents are subtracted.