The multiplication property of exponents is a fundamental rule in algebra. When we multiply two exponential expressions that have the same base, we keep the base and add the exponents together. For example, 2 to the 3rd power times 2 to the 4th power equals 2 to the 7th power.
Let's work through a step-by-step example. We start with 3 squared times 3 to the fifth power. First, we identify that both terms have the same base, which is 3. Next, we keep the base unchanged and add the exponents: 2 plus 5 equals 7. Finally, we can calculate that 3 to the seventh power equals 2187.
To understand this property visually, remember that exponents represent repeated multiplication. 2 to the 3rd power means 2 times 2 times 2, and 2 to the 4th power means 2 times 2 times 2 times 2. When we multiply these together, we get seven factors of 2, which equals 2 to the 7th power.
Let's practice with more examples to reinforce our understanding. X to the 3rd times X squared equals X to the 5th. Five to the 1st times five to the 6th equals five to the 7th. Y to the 4th times Y cubed equals Y to the 7th. And ten squared times ten cubed equals ten to the 5th. Notice how we always keep the same base and add the exponents.
Let's summarize the key points about the multiplication property of exponents. First, the bases must be identical for this rule to apply. Second, we keep the base unchanged and simply add the exponents. Remember, this property only works for multiplication, not addition. For example, 3 squared times 3 to the fourth equals 3 to the sixth, but 2 cubed times 3 squared does not equal 6 to the fifth power.