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 511 as the product of two powers. Explain why all three of your expressions are equal to 511.
5 answers
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.
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.
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 $5^{11}$.
1 answer