10.
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
1. (5^5) * (5^6) - This is equal to 5 raised to the 5th power multiplied by 5 raised to the 6th power, which equals 5 raised to the 11th power because when you multiply two powers with the same base, you add their exponents.
2. (5^10) * (5^1) - This is equal to 5 raised to the 10th power multiplied by 5 raised to the 1st power, which also equals 5 raised to the 11th power because, as mentioned before, adding the exponents of two powers with the same base gives you the result of multiplying those two powers.
3. (5^3) * (5^3) * (5^3) * (5^2) - This is equal to 5 raised to the 3rd power multiplied by itself three times, and then multiplied by 5 raised to the 2nd power. This equals 5 raised to the 11th power because when you multiply multiple powers with the same base, you add their exponents, which in this case would be 3+3+3+2=11.
All three expressions are equivalent to 5^11 because they are just different ways of expressing the same value. By using the laws of exponents, which state that multiplying two powers with the same base is equivalent to adding their exponents, we can see that all three expressions result in multiplying different combinations of factors that when combined, equal 5 raised to the 11th power.
Three different ways to write 5^11 as the product of two powers are:
1. (5^5) * (5^6): In this expression, we are multiplying 5 raised to the power of 5 with 5 raised to the power of 6. Since the base, which is 5, is common in both powers, we can add their exponents, which gives us 11. Therefore, (5^5) * (5^6) is equal to 5^11.
2. (5^2) * (5^9): In this expression, we are multiplying 5 raised to the power of 2 with 5 raised to the power of 9. Again, since the base is the same, we add the exponents, which equals 11. Hence, (5^2) * (5^9) is equal to 5^11.
3. (5^3) * (5^8): Here, we are multiplying 5 raised to the power of 3 with 5 raised to the power of 8. By adding the exponents, we get 11. Thus, (5^3) * (5^8) is also equal to 5^11.
In all three expressions, the powers are multiplied together to represent 5 raised to the power of 11. The rules of exponentiation state that when the base is the same, the exponents can be added when multiplying the powers. Hence, all three expressions are equal to 5^11.
To write 5^11 as the product of two powers, we need to express it as 5 raised to a power multiplied by another power. Here are three different ways to do that:
1. 5^11 = (5^5) * (5^6)
Explanation: We can break down 5^11 as the product of two powers, where the first power is 5^5 and the second power is 5^6. This is equal to 5 raised to the sum of the exponents, which is 5^11.
2. 5^11 = (5^2) * (5^9)
Explanation: Another way to express 5^11 as the product of two powers is by using 5^2 and 5^9. When we multiply these two powers together, we get 5^11, since the exponents are added.
3. 5^11 = (5^3) * (5^8)
Explanation: Finally, we can write 5^11 as the product of two powers using 5^3 and 5^8. When we multiply these two powers, we get 5^11, again because we add the exponents.
All three expressions are equal to 5^11 because they follow the exponentiation rules. When we multiply two powers with the same base, we add their exponents. In each of these cases, the exponents add up to 11, hence resulting in 5^11.