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To factor the expression \(12y + 6\) by finding the greatest common factor (GCF), follow these steps:
Identify the GCF: The GCF of the coefficients 12 and 6 is 6.
Factor out the GCF: Divide each term by the GCF and write the expression in factored form:
\[ 12y + 6 = 6(2y + 1) \]
Thus, the factored form of \(12y + 6\) is \(6(2y + 1)\).
To factor the expression \(12y + 6\) using the greatest common factor (GCF), follow these steps:
Identify the GCF: The GCF of the coefficients 12 and 6 is 6.
Factor out the GCF: Divide each term by the GCF (6) and write the expression in factored form:
\[ 12y + 6 = 6(2y) + 6(1) = 6(2y + 1) \]
So, the factored form of \(12y + 6\) is:
\[ 6(2y + 1) \]
Certainly! Here's a step-by-step breakdown of factoring the expression \(12y + 6\) using the greatest common factor (GCF):
The expression is: \[ 12y + 6 \] The coefficients are \(12\) (from \(12y\)) and \(6\).
To find the GCF of \(12\) and \(6\), we can list the factors:
The greatest common factor is \(6\).
Now that we know the GCF is \(6\), we will factor \(6\) out of each term in the expression \(12y + 6\).
Rewrite each term as follows:
Now we can express the original expression in terms of the GCF: \[ 12y + 6 = 6(2y) + 6(1) \]
Now, factor \(6\) out of the entire expression: \[ 12y + 6 = 6(2y + 1) \]
Thus, the fully factored form of \(12y + 6\) is: \[ 6(2y + 1) \]
This shows all the steps taken to factor the expression using the GCF.
Let’s break down how we arrived at \(2y\) and \(1\) when factoring \(12y + 6\) with the greatest common factor (GCF) of \(6\).
\[ 12y + 6 \]
As previously mentioned, the GCF of \(12\) and \(6\) is \(6\).
For the first term \(12y\):
For the second term \(6\):
Now, we can rewrite the original expression \(12y + 6\) by factoring out the GCF:
\[ 12y + 6 = 6 \cdot 2y + 6 \cdot 1 \] This allows us to factor out the \(6\):
\[ 12y + 6 = 6(2y + 1) \]
So, when factoring \(12y + 6\):
This shows the reasoning behind arriving at \(2y\) and \(1\).
To simplify and add the expression \( 0.5k^{-1} + 0.4^{-k} \), follow these steps:
Rewrite the expression: \[ 205(0.5k^{-1}) + 2(0.4^{-k}) \]
Distribute the constants: \[ = 205 \cdot 0.5k^{-1} + 2 \cdot 0.4^{-k} \]
Calculate the products:
For the first term: \[ 205 \cdot 0.5 = 102.5 \] Thus, the first term becomes: \[ 102.5k^{-1} = \frac{102.5}{k} \]
For the second term: \[ 2 \cdot 0.4^{-k} = 2 \cdot \frac{1}{0.4^k} \]
Combine both parts: The final result is: \[ \frac{102.5}{k} + \frac{2}{0.4^k} \]
Since \(0.4 = \frac{2}{5}\), you can express \(0.4^k\) as: \[ 0.4^k = \left( \frac{2}{5} \right)^k = \frac{2^k}{5^k} \] Therefore: \[ \frac{2}{0.4^k} = \frac{2}{\frac{2^k}{5^k}} = \frac{2 \cdot 5^k}{2^k} \]
If you have a specific value for \(k\), substitute it to get a numerical result. Otherwise, this is the simplified form of the expression.
Certainly! Let’s simplify the expression step by step.
Given the expression:
\[ 205(0.5k^{-1}) + 2(0.4^{-k}) \]
For the first term: \[ 205 \cdot 0.5k^{-1} = 102.5k^{-1} = \frac{102.5}{k} \]
For the second term: \[ 2(0.4^{-k}) = 2 \cdot \frac{1}{0.4^k} = \frac{2}{0.4^k} \]
Since \(0.4\) can be written as \(\frac{2}{5}\), we can express \(0.4^k\) as: \[ 0.4^k = \left(\frac{2}{5}\right)^k = \frac{2^k}{5^k} \]
This means: \[ \frac{2}{0.4^k} = \frac{2}{\frac{2^k}{5^k}} = \frac{2 \cdot 5^k}{2^k} \]
Now we put both simplified expressions back together: \[ \frac{102.5}{k} + \frac{2 \cdot 5^k}{2^k} \]
So, the final expression simplified is: \[ \frac{102.5}{k} + \frac{2 \cdot 5^k}{2^k} \]
If you want to combine these into a single fraction, it would look like this, but remember that they can only be combined if they have a common denominator:
The common denominator would be \(k \cdot 2^k\): \[ = \frac{102.5 \cdot 2^k + 2 \cdot 5^k \cdot k}{k \cdot 2^k} \]
The final simplified expression is: \[ \frac{102.5 \cdot 2^k + 2 \cdot 5^k \cdot k}{k \cdot 2^k} \]
Feel free to provide a value for \(k\) if you need further evaluation!
