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Factor 12y+6 using the GCF
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Answers
Answered by
GPT-4o mini
To factor the expression \(12y + 6\) by finding the greatest common factor (GCF), follow these steps:
1. **Identify the GCF**: The GCF of the coefficients 12 and 6 is 6.
2. **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 \(6(2y + 1)\).
1. **Identify the GCF**: The GCF of the coefficients 12 and 6 is 6.
2. **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 \(6(2y + 1)\).
Answered by
GPT-4o mini
To factor the expression \(12y + 6\) using the greatest common factor (GCF), follow these steps:
1. **Identify the GCF**: The GCF of the coefficients 12 and 6 is 6.
2. **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:
\[
6(2y + 1)
\]
1. **Identify the GCF**: The GCF of the coefficients 12 and 6 is 6.
2. **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:
\[
6(2y + 1)
\]
Answered by
GPT-4o mini
Certainly! Here's a step-by-step breakdown of factoring the expression \(12y + 6\) using the greatest common factor (GCF):
### Step 1: Identify the coefficients
The expression is:
\[
12y + 6
\]
The coefficients are \(12\) (from \(12y\)) and \(6\).
### Step 2: Find the GCF of the coefficients
To find the GCF of \(12\) and \(6\), we can list the factors:
- Factors of \(12\): \(1, 2, 3, 4, 6, 12\)
- Factors of \(6\): \(1, 2, 3, 6\)
The greatest common factor is \(6\).
### Step 3: Factor out the GCF
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:**
- The first term: \(12y = 6 \cdot 2y\)
- The second term: \(6 = 6 \cdot 1\)
### Step 4: Write the expression in factored form
Now we can express the original expression in terms of the GCF:
\[
12y + 6 = 6(2y) + 6(1)
\]
### Step 5: Combine the terms inside the parentheses
Now, factor \(6\) out of the entire expression:
\[
12y + 6 = 6(2y + 1)
\]
### Conclusion
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.
### Step 1: Identify the coefficients
The expression is:
\[
12y + 6
\]
The coefficients are \(12\) (from \(12y\)) and \(6\).
### Step 2: Find the GCF of the coefficients
To find the GCF of \(12\) and \(6\), we can list the factors:
- Factors of \(12\): \(1, 2, 3, 4, 6, 12\)
- Factors of \(6\): \(1, 2, 3, 6\)
The greatest common factor is \(6\).
### Step 3: Factor out the GCF
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:**
- The first term: \(12y = 6 \cdot 2y\)
- The second term: \(6 = 6 \cdot 1\)
### Step 4: Write the expression in factored form
Now we can express the original expression in terms of the GCF:
\[
12y + 6 = 6(2y) + 6(1)
\]
### Step 5: Combine the terms inside the parentheses
Now, factor \(6\) out of the entire expression:
\[
12y + 6 = 6(2y + 1)
\]
### Conclusion
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.
Answered by
GPT-4o mini
Let’s break down how we arrived at \(2y\) and \(1\) when factoring \(12y + 6\) with the greatest common factor (GCF) of \(6\).
### Given Expression:
\[
12y + 6
\]
### Step 1: Identify the GCF
As previously mentioned, the GCF of \(12\) and \(6\) is \(6\).
### Step 2: Rewrite each term as a product of the GCF
1. For the first term \(12y\):
- We divide \(12y\) by the GCF (which is \(6\)):
\[
\frac{12y}{6} = 2y
\]
Therefore, we can express \(12y\) as:
\[
12y = 6 \cdot 2y
\]
2. For the second term \(6\):
- We divide \(6\) by the GCF \(6\):
\[
\frac{6}{6} = 1
\]
Hence, we can express \(6\) as:
\[
6 = 6 \cdot 1
\]
### Step 3: Combine the results
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 \(6\):
\[
12y + 6 = 6(2y + 1)
\]
### Summary
So, when factoring \(12y + 6\):
- We factored out \(6\) to arrive at \(2y\) from \(12y\) and \(1\) from \(6\), leading to the final factored form:
\[
6(2y + 1)
\]
This shows the reasoning behind arriving at \(2y\) and \(1\).
### Given Expression:
\[
12y + 6
\]
### Step 1: Identify the GCF
As previously mentioned, the GCF of \(12\) and \(6\) is \(6\).
### Step 2: Rewrite each term as a product of the GCF
1. For the first term \(12y\):
- We divide \(12y\) by the GCF (which is \(6\)):
\[
\frac{12y}{6} = 2y
\]
Therefore, we can express \(12y\) as:
\[
12y = 6 \cdot 2y
\]
2. For the second term \(6\):
- We divide \(6\) by the GCF \(6\):
\[
\frac{6}{6} = 1
\]
Hence, we can express \(6\) as:
\[
6 = 6 \cdot 1
\]
### Step 3: Combine the results
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 \(6\):
\[
12y + 6 = 6(2y + 1)
\]
### Summary
So, when factoring \(12y + 6\):
- We factored out \(6\) to arrive at \(2y\) from \(12y\) and \(1\) from \(6\), leading to the final factored form:
\[
6(2y + 1)
\]
This shows the reasoning behind arriving at \(2y\) and \(1\).
Answered by
GPT-4o mini
To simplify and add the expression \( 0.5k^{-1} + 0.4^{-k} \), follow these steps:
1. **Rewrite the expression**:
\[
205(0.5k^{-1}) + 2(0.4^{-k})
\]
2. **Distribute the constants**:
\[
= 205 \cdot 0.5k^{-1} + 2 \cdot 0.4^{-k}
\]
3. **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:
\[
2 \cdot 0.4^{-k} = 2 \cdot \frac{1}{0.4^k}
\]
4. **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:
\[
\frac{2}{0.4^k} = \frac{2}{\frac{2^k}{5^k}} = \frac{2 \cdot 5^k}{2^k}
\]
5. **Final expression**:
So, the final expression in decimal form is approximately:
\[
\frac{102.5}{k} + \frac{2 \cdot 5^k}{2^k}
\]
If you have a specific value for \(k\), substitute it to get a numerical result. Otherwise, this is the simplified form of the expression.
1. **Rewrite the expression**:
\[
205(0.5k^{-1}) + 2(0.4^{-k})
\]
2. **Distribute the constants**:
\[
= 205 \cdot 0.5k^{-1} + 2 \cdot 0.4^{-k}
\]
3. **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:
\[
2 \cdot 0.4^{-k} = 2 \cdot \frac{1}{0.4^k}
\]
4. **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:
\[
\frac{2}{0.4^k} = \frac{2}{\frac{2^k}{5^k}} = \frac{2 \cdot 5^k}{2^k}
\]
5. **Final expression**:
So, the final expression in decimal form is approximately:
\[
\frac{102.5}{k} + \frac{2 \cdot 5^k}{2^k}
\]
If you have a specific value for \(k\), substitute it to get a numerical result. Otherwise, this is the simplified form of the expression.
Answered by
GPT-4o mini
Certainly! Let’s simplify the expression step by step.
Given the expression:
\[
205(0.5k^{-1}) + 2(0.4^{-k})
\]
### Step 1: Distribute the constants
1. **For the first term**:
\[
205 \cdot 0.5k^{-1} = 102.5k^{-1} = \frac{102.5}{k}
\]
2. **For the second term**:
\[
2(0.4^{-k}) = 2 \cdot \frac{1}{0.4^k} = \frac{2}{0.4^k}
\]
### Step 2: Rewrite using exponent properties
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:
\[
\frac{2}{0.4^k} = \frac{2}{\frac{2^k}{5^k}} = \frac{2 \cdot 5^k}{2^k}
\]
### Step 3: Combine the terms
Now we put both simplified expressions back together:
\[
\frac{102.5}{k} + \frac{2 \cdot 5^k}{2^k}
\]
### Final Expression
So, the final expression simplified is:
\[
\frac{102.5}{k} + \frac{2 \cdot 5^k}{2^k}
\]
### Additional Simplification (if needed)
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!
Given the expression:
\[
205(0.5k^{-1}) + 2(0.4^{-k})
\]
### Step 1: Distribute the constants
1. **For the first term**:
\[
205 \cdot 0.5k^{-1} = 102.5k^{-1} = \frac{102.5}{k}
\]
2. **For the second term**:
\[
2(0.4^{-k}) = 2 \cdot \frac{1}{0.4^k} = \frac{2}{0.4^k}
\]
### Step 2: Rewrite using exponent properties
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:
\[
\frac{2}{0.4^k} = \frac{2}{\frac{2^k}{5^k}} = \frac{2 \cdot 5^k}{2^k}
\]
### Step 3: Combine the terms
Now we put both simplified expressions back together:
\[
\frac{102.5}{k} + \frac{2 \cdot 5^k}{2^k}
\]
### Final Expression
So, the final expression simplified is:
\[
\frac{102.5}{k} + \frac{2 \cdot 5^k}{2^k}
\]
### Additional Simplification (if needed)
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!