List three different ways to write 5^12 as the product of two powers. Explain why all three of your expressions are equal to 5^12.
To write 5^12 as the product of two powers, we need to find pairs of exponents that when multiplied together equal 12. Here are three different ways to write 5^12 as the product of two powers:
1. 5^6 * 5^6: Here, we take advantage of the property of exponents that states when two powers with the same base are multiplied, we add their exponents. So, 5^6 * 5^6 = 5^(6+6) = 5^12.
2. 5^4 * 5^8: In this case, we distribute the exponent 12 into two parts in a different way. Again, using the exponent property, when two powers with the same base are multiplied, we add their exponents. So, 5^4 * 5^8 = 5^(4+8) = 5^12.
3. 5^2 * 5^10: Here, we distribute the exponent 12 into two parts differently from the previous examples. And as before, when two powers with the same base are multiplied, we add their exponents. So, 5^2 * 5^10 = 5^(2+10) = 5^12.
All three expressions are equal to 5^12 because they all follow the exponent properties that state the sum of the exponents when multiplying powers with the same base should be the same. In each case, the base 5 remains the same, and the total exponent is 12. Therefore, all three expressions yield the same result, which is 5^12.