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.
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1. 5^5 * 5^6 = 5^11
- Here, we are using the rule that when multiplying powers with the same base, we add their exponents.
2. 5^2 * (5^3)^2 = 5^2 * 5^6 = 5^8 * 5^3 = 5^11
- Here, we are using the rule that when raising a power to another power, we multiply their exponents.
3. (25)^5/2 = (5^2)^5/2 = 5^5 * 5^5 = 5^11
- Here, we are using the rule that when taking the square root of a power, we divide the exponent by 2.
All three expressions are equal to 5^11 because they all represent the same base (5) being multiplied by itself 11 times, which is what 5^11 means by definition. We are just using different mathematical rules to manipulate the exponents and rewrite the expression in a different way.
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To list three different ways to write 5^11 as the product of two powers, we can use exponent rules:
1. 5^11 = 5^10 * 5^1
2. 5^11 = 5^5 * 5^6
3. 5^11 = 5^7 * 5^4
To understand why all three expressions are equal to 5^11, we need to review the rules of exponents. The rule for multiplying powers with the same base states that when you multiply two numbers with the same base, you can add their exponents. Therefore, in all three expressions, we are multiplying two powers of 5.
In the first expression (5^10 * 5^1), we have two powers of 5 being multiplied: 5^10 and 5^1. Adding their exponents gives us 10 + 1 = 11, so we get 5^11.
In the second expression (5^5 * 5^6), the exponents are 5 and 6. Adding them together gives us 5 + 6 = 11, resulting in 5^11.
Finally, in the third expression (5^7 * 5^4), the exponents are 7 and 4. Adding them gives us 7 + 4 = 11, which equals 5^11.
Therefore, all three expressions represent the same value, which is 5^11.