List three different ways to write 511 as the product of two powers. Explain why all three of your expressions are equal to 511.
1. $511 = 2^9 \cdot 1^1$
2. $511 = 7^1 \cdot 73^1$
3. $511 = 11^1 \cdot 47^1$
All three expressions are equal to 511 because they are all just different ways to decompose 511 into a product of its prime factors.
In the first expression, we are decomposing 511 as $2^9$ times $1^1$, which means we are essentially saying that 511 has 9 twos in its prime factorization and 1 one (which doesn't really affect anything).
In the second expression, we are decomposing 511 as $7^1$ times $73^1$, which means we are saying that 511 has a prime factor of 7 and a prime factor of 73.
In the third expression, we are decomposing 511 as $11^1$ times $47^1$, which means we are saying that 511 has a prime factor of 11 and a prime factor of 47.
So while the expressions may look different, they all ultimately represent the same number: 511.
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. $5^{11} = 5^5 \cdot 5^6$
2. $5^{11} = (5^2)^5 \cdot 5^1$
3. $5^{11} = (5^3)^3$
All three expressions are equal to $5^{11}$ because they are all just different ways to decompose $5^{11}$ into a product of powers of 5.
In the first expression, we're using the fact that we can multiply powers with the same base by adding their exponents. So we're breaking up $5^{11}$ into $5^5$ and $5^6$, whose exponents add up to 11.
In the second expression, we're using the fact that we can use the power of a power rule, which states that to raise a power to another power, we multiply the exponents. So we're breaking up $5^{11}$ into $(5^2)^5$, which we simplify using the power of a power rule to $5^{10}$, and $5^1$.
In the third expression, we're using the fact that we can use the power of a power rule again to simplify $5^{11}$ into $(5^3)^3$, which is just $5^9$ raised to the power of 3.
1. $5^{11} = 5^5 \cdot 5^6$
2. $5^{11} = (5^2)^5 \cdot 5^1$
3. $5^{11} = (5^3)^3$
In three sentences Explain why all three of your expressions are equal to 5^11.
All three expressions represent different ways of breaking up 5 raised to the power of 11 into a product of powers. They are all based on the rules of exponents, namely the power of a product rule and the power of a power rule. Regardless of the specific decomposition used, they all ultimately simplify to the same value of $5^{11}$.
To write 511 as the product of two powers, we need to find two numbers whose product is 511. There are several methods to approach this problem. Here are three different ways:
Method 1: Prime Factorization
1. Start by finding the prime factorization of 511: 511 = 7 * 73.
2. Express 511 as a product of two powers: 511 = 7^1 * 73^1.
Method 2: Trial and Error
1. Start by listing powers of numbers and looking for a combination that gives 511.
- 2^1 = 2, 2^2 = 4, 2^3 = 8, 2^4 = 16, 2^5 = 32, 2^6 = 64, 2^7 = 128, 2^8 = 256, 2^9 = 512.
- 3^1 = 3, 3^2 = 9, 3^3 = 27, 3^4 = 81, 3^5 = 243.
- 4^1 = 4, 4^2 = 16, 4^3 = 64, 4^4 = 256.
2. By trial and error, we find that 7^1 * 73^1 = 511.
Method 3: Square Root
1. Take the square root of 511 using a calculator or approximation methods: √511 ≈ 22.6.
2. Look for factors of 511 that are close to the square root.
- 22 * 23 = 506, which is close but not equal to 511.
3. Try the next closest numbers and find: 23 * 23 = 529, which is greater than 511.
4. Therefore, we can express 511 as 23^2 - 12^2, which leads to the difference of squares formula: 511 = (23 + 12)(23 - 12).
5. Simplifying, we get 511 = 35 * 11 = 7^1 * 73^1.
Explanation of Equality:
In all three expressions, 511 is equal to the product of two powers, namely 7 and 73. The first method, prime factorization, breaks down 511 into its prime factors, which gives us its fundamental composition. The second method, trial and error, explores various powers of numbers until finding the correct combination that equals 511. The third method, using the square root and difference of squares, provides an alternate way to obtain the prime factors of 511. Regardless of the method used, the resulting expressions are equal to 511 because they all represent the same fundamental composition of the number.