List three different ways to write 5^11 as the product of two powers. Explain why all three of your expressions are equal to 5^11.
There are many ways to write 5^11 as the product of two powers, but three possible ways are:
1. 5^11 = 5^10 × 5^1
2. 5^11 = 5^6 × 5^5
3. 5^11 = 5^3 × 5^8
To see why these expressions are equal to 5^11, we can use the exponent rules:
1. When we multiply powers with the same base, we add their exponents. So 5^10 × 5^1 = 5^(10+1) = 5^11.
2. Similarly, 5^6 × 5^5 = 5^(6+5) = 5^11.
3. Finally, 5^3 × 5^8 = 5^(3+8) = 5^11.
In each case, we are simply adding the exponents of two powers with the same base, and the result is always 5^11.