List three different ways to write 5 to the 11th power as a product of two powers
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To list three different ways to write 5 to the 11th power as a product of two powers, we can use some properties of exponents.
1. By dividing the exponent: We can divide the exponent 11 into two powers, such as 5 to the 5th power times 5 to the 6th power. This can be written as (5^5) * (5^6).
2. By multiplying the base: Another way is to express 5 to the 11th power as the product of two bases with the exponent of half the original power. In this case, it can be written as (5^6) * (5^5).
3. By using negative exponents: We can use the property that a negative exponent is equal to the reciprocal of the positive exponent. So, 5 to the 11th power can be written as (5^(-5)) * (5^16).
By applying these methods, we have three different ways to write 5 to the 11th power as a product of two powers: (5^5) * (5^6), (5^6) * (5^5), and (5^(-5)) * (5^16).
To write 5 to the 11th power as a product of two powers, we can use the properties of exponents. Here are three different ways to express it:
1. 5^11 = (5^5) * (5^6)
In this case, we split the power into two parts, using the fact that 11 = 5 + 6.
2. 5^11 = (5^9) * (5^2)
Here, we split the power into 9 and 2, recognizing that 11 = 9 + 2.
3. 5^11 = (5^3) * (5^8)
In this method, we split the power into 3 and 8, which equals 11.
Note: In all cases, the sum of the exponents in the two powers adds up to 11, which represents the original exponent.