What are 2 different way to write 3^7 as the product of two powers
1) 3^7 can be written as (3^4) * (3^3)
2) 3^7 can also be written as (3^6) * 3
Explain why both of the expressions are equal to 3^7
Both of the expressions are equal to 3^7 because multiplication of powers follows the rule that when multiplying powers with the same base, you add the exponents.
In the first expression, (3^4) * (3^3), the base 3 is the same in both terms. Adding the exponents gives us 4 + 3 = 7, so the expression simplifies to 3^7.
In the second expression, (3^6) * 3, again we have the same base 3. The exponent 6 multiplied by 1 is 6, so the expression simplifies to 3^7.
To write 3^7 as the product of two powers, we need to find two powers that when multiplied together, equal 3^7.
Method 1:
We can break down 3^7 into two powers by grouping the factors together. For example, we can write it as (3^4) * (3^3). This is because 3^4 = 81 and 3^3 = 27, and when multiplied together, 81 * 27 = 2187, which is equal to 3^7.
Method 2:
We can also break down 3^7 into two powers by using exponent properties. Since 7 is an odd number, we can find the product of two powers by dividing the exponent by 2 and distributing it between the two powers. In this case, we can write it as (3^3) * (3^4). This is because 7 divided by 2 is 3 remainder 1. Therefore, we can divide the exponent 7 into two powers, where one power has an exponent of 3 (3^3) and the other power has an exponent of the remaining 4 (3^4). When multiplied together, 3^3 * 3^4 equals 27 * 81, which is again equal to 2187, or 3^7.