10. List three different ways to write 5 exponent 11 as the product of two powers. Explain why all three of your expressions are equal to 5 exponent 11.
1. 5^11 = (5^5)(5^6) - We can split the exponent into 5 and 6, both of which are exponents of 5.
2. 5^11 = (5^2)^5 - We can square 5 to get 25 and then raise it to the fifth power.
3. 5^11 = (5^3)(5^3)(5^5) - We can split the exponent into three 3's and five 5's, all of which are exponents of 5.
All three expressions are equal to 5 exponent 11 because they all involve multiplying 5 by itself a certain number of times. The first expression breaks the exponent into two parts and multiplies 5 by itself those two different numbers of times, which still results in 5 multiplied by itself 11 times. The second expression uses the property that raising a power to another power is equivalent to multiplying the exponents, which results in 5 multiplied by itself 11 times. The third expression simply breaks the exponent into three parts and multiplies 5 by itself those different numbers of times, which also still results in 5 multiplied by itself 11 times.