Question
What is the quotient of the following expression? State the answer in simplest radical form. Rationalize the denominator if necessary. 12√24/3√300 (1 point)
Answers
GPT-4o mini
To simplify the expression \(\frac{12\sqrt{24}}{3\sqrt{300}}\), we can start by simplifying the square roots and then simplifying the fraction.
1. **Simplify the square roots**:
\(\sqrt{24}\) can be simplified as follows:
\[
\sqrt{24} = \sqrt{4 \cdot 6} = \sqrt{4} \cdot \sqrt{6} = 2\sqrt{6}
\]
Therefore, we replace \(\sqrt{24}\) in the expression:
\[
12\sqrt{24} = 12(2\sqrt{6}) = 24\sqrt{6}
\]
Next, we simplify \(\sqrt{300}\):
\[
\sqrt{300} = \sqrt{100 \cdot 3} = \sqrt{100} \cdot \sqrt{3} = 10\sqrt{3}
\]
Thus we replace \(\sqrt{300}\):
\[
3\sqrt{300} = 3(10\sqrt{3}) = 30\sqrt{3}
\]
2. **Substitute back into the expression**:
Now we have:
\[
\frac{12\sqrt{24}}{3\sqrt{300}} = \frac{24\sqrt{6}}{30\sqrt{3}}
\]
3. **Simplify the fraction**:
We can simplify the coefficients \(24\) and \(30\) by finding the greatest common divisor (GCD), which is \(6\):
\[
\frac{24}{30} = \frac{24 \div 6}{30 \div 6} = \frac{4}{5}
\]
Therefore, our expression simplifies to:
\[
\frac{4}{5} \cdot \frac{\sqrt{6}}{\sqrt{3}} = \frac{4}{5} \cdot \sqrt{\frac{6}{3}} = \frac{4}{5} \cdot \sqrt{2}
\]
4. **Combine the terms**:
Thus, we can write our final result as:
\[
\frac{4\sqrt{2}}{5}
\]
The final answer in simplest radical form is:
\[
\frac{4\sqrt{2}}{5}
\]
1. **Simplify the square roots**:
\(\sqrt{24}\) can be simplified as follows:
\[
\sqrt{24} = \sqrt{4 \cdot 6} = \sqrt{4} \cdot \sqrt{6} = 2\sqrt{6}
\]
Therefore, we replace \(\sqrt{24}\) in the expression:
\[
12\sqrt{24} = 12(2\sqrt{6}) = 24\sqrt{6}
\]
Next, we simplify \(\sqrt{300}\):
\[
\sqrt{300} = \sqrt{100 \cdot 3} = \sqrt{100} \cdot \sqrt{3} = 10\sqrt{3}
\]
Thus we replace \(\sqrt{300}\):
\[
3\sqrt{300} = 3(10\sqrt{3}) = 30\sqrt{3}
\]
2. **Substitute back into the expression**:
Now we have:
\[
\frac{12\sqrt{24}}{3\sqrt{300}} = \frac{24\sqrt{6}}{30\sqrt{3}}
\]
3. **Simplify the fraction**:
We can simplify the coefficients \(24\) and \(30\) by finding the greatest common divisor (GCD), which is \(6\):
\[
\frac{24}{30} = \frac{24 \div 6}{30 \div 6} = \frac{4}{5}
\]
Therefore, our expression simplifies to:
\[
\frac{4}{5} \cdot \frac{\sqrt{6}}{\sqrt{3}} = \frac{4}{5} \cdot \sqrt{\frac{6}{3}} = \frac{4}{5} \cdot \sqrt{2}
\]
4. **Combine the terms**:
Thus, we can write our final result as:
\[
\frac{4\sqrt{2}}{5}
\]
The final answer in simplest radical form is:
\[
\frac{4\sqrt{2}}{5}
\]
incorrect.
GPT-4o mini
Let’s go through the calculations carefully step by step again.
We need to simplify the expression \(\frac{12\sqrt{24}}{3\sqrt{300}}\).
1. **Simplify the square roots**:
- First, simplify \(\sqrt{24}\):
\[
\sqrt{24} = \sqrt{4 \cdot 6} = \sqrt{4}\sqrt{6} = 2\sqrt{6}
\]
- Next, simplify \(\sqrt{300}\):
\[
\sqrt{300} = \sqrt{100 \cdot 3} = \sqrt{100}\sqrt{3} = 10\sqrt{3}
\]
2. **Substituting back into the expression**:
Now substitute back into the original expression:
\[
\frac{12\sqrt{24}}{3\sqrt{300}} = \frac{12(2\sqrt{6})}{3(10\sqrt{3})} = \frac{24\sqrt{6}}{30\sqrt{3}}
\]
3. **Simplifying the fraction**:
We can simplify the coefficients \(24\) and \(30\):
\[
\frac{24}{30} = \frac{4}{5}
\]
Next, we are left with:
\[
\frac{4\sqrt{6}}{5\sqrt{3}}
\]
4. **Rationalizing the denominator**:
To rationalize the denominator, multiply the numerator and denominator by \(\sqrt{3}\):
\[
\frac{4\sqrt{6}}{5\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{4\sqrt{6} \cdot \sqrt{3}}{5 \cdot 3} = \frac{4\sqrt{18}}{15}
\]
Now simplify \(\sqrt{18}\):
\[
\sqrt{18} = \sqrt{9 \cdot 2} = 3\sqrt{2}
\]
Thus, we have:
\[
\frac{4 \cdot 3\sqrt{2}}{15} = \frac{12\sqrt{2}}{15}
\]
We can simplify this further:
\[
\frac{12}{15} = \frac{4}{5}
\]
Finally, the expression in simplest radical form is:
\[
\frac{4\sqrt{2}}{5}
\]
Thank you for your patience, and I apologize for any confusion earlier!
We need to simplify the expression \(\frac{12\sqrt{24}}{3\sqrt{300}}\).
1. **Simplify the square roots**:
- First, simplify \(\sqrt{24}\):
\[
\sqrt{24} = \sqrt{4 \cdot 6} = \sqrt{4}\sqrt{6} = 2\sqrt{6}
\]
- Next, simplify \(\sqrt{300}\):
\[
\sqrt{300} = \sqrt{100 \cdot 3} = \sqrt{100}\sqrt{3} = 10\sqrt{3}
\]
2. **Substituting back into the expression**:
Now substitute back into the original expression:
\[
\frac{12\sqrt{24}}{3\sqrt{300}} = \frac{12(2\sqrt{6})}{3(10\sqrt{3})} = \frac{24\sqrt{6}}{30\sqrt{3}}
\]
3. **Simplifying the fraction**:
We can simplify the coefficients \(24\) and \(30\):
\[
\frac{24}{30} = \frac{4}{5}
\]
Next, we are left with:
\[
\frac{4\sqrt{6}}{5\sqrt{3}}
\]
4. **Rationalizing the denominator**:
To rationalize the denominator, multiply the numerator and denominator by \(\sqrt{3}\):
\[
\frac{4\sqrt{6}}{5\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{4\sqrt{6} \cdot \sqrt{3}}{5 \cdot 3} = \frac{4\sqrt{18}}{15}
\]
Now simplify \(\sqrt{18}\):
\[
\sqrt{18} = \sqrt{9 \cdot 2} = 3\sqrt{2}
\]
Thus, we have:
\[
\frac{4 \cdot 3\sqrt{2}}{15} = \frac{12\sqrt{2}}{15}
\]
We can simplify this further:
\[
\frac{12}{15} = \frac{4}{5}
\]
Finally, the expression in simplest radical form is:
\[
\frac{4\sqrt{2}}{5}
\]
Thank you for your patience, and I apologize for any confusion earlier!
answer is wrong make 1000000000% sure your correct
GPT-4o mini
Let’s carefully go through the entire process once again step-by-step to simplify \(\frac{12\sqrt{24}}{3\sqrt{300}}\) and confirm the solution thoroughly.
### Step 1: Simplify the Square Roots
1. **Calculate \(\sqrt{24}\)**:
\[
\sqrt{24} = \sqrt{4 \cdot 6} = \sqrt{4}\sqrt{6} = 2\sqrt{6}
\]
2. **Calculate \(\sqrt{300}\)**:
\[
\sqrt{300} = \sqrt{100 \cdot 3} = \sqrt{100}\sqrt{3} = 10\sqrt{3}
\]
### Step 2: Substitute Back into the Expression
We can now substitute these back into the original expression:
\[
\frac{12\sqrt{24}}{3\sqrt{300}} = \frac{12(2\sqrt{6})}{3(10\sqrt{3})} = \frac{24\sqrt{6}}{30\sqrt{3}}
\]
### Step 3: Simplify the Coefficient Fraction
Next, simplify the coefficients:
\[
\frac{24}{30} = \frac{4}{5}
\]
So we have:
\[
\frac{4\sqrt{6}}{5\sqrt{3}}
\]
### Step 4: Rationalize the Denominator
Now we need to rationalize:
\[
\frac{4\sqrt{6}}{5\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{4\sqrt{6} \cdot \sqrt{3}}{5 \cdot 3} = \frac{4\sqrt{18}}{15}
\]
### Step 5: Simplify \(\sqrt{18}\)
Now simplify:
\[
\sqrt{18} = \sqrt{9 \cdot 2} = 3\sqrt{2}
\]
### Step 6: Substitute Back
Now substitute it back:
\[
\frac{4(3\sqrt{2})}{15} = \frac{12\sqrt{2}}{15}
\]
### Step 7: Final Simplification
Finally, simplify \(\frac{12}{15}\):
\[
\frac{12}{15} = \frac{4}{5}
\]
So the final expression becomes:
\[
\frac{4\sqrt{2}}{5}
\]
### Conclusion
After carefully going through the calculations again, I confirm that the answer is indeed:
\(\frac{4\sqrt{2}}{5}\).
If there is a specific aspect of this problem that seems incorrect or any specific point of confusion, please let me know and I will be happy to clarify further. Thank you for your patience!
### Step 1: Simplify the Square Roots
1. **Calculate \(\sqrt{24}\)**:
\[
\sqrt{24} = \sqrt{4 \cdot 6} = \sqrt{4}\sqrt{6} = 2\sqrt{6}
\]
2. **Calculate \(\sqrt{300}\)**:
\[
\sqrt{300} = \sqrt{100 \cdot 3} = \sqrt{100}\sqrt{3} = 10\sqrt{3}
\]
### Step 2: Substitute Back into the Expression
We can now substitute these back into the original expression:
\[
\frac{12\sqrt{24}}{3\sqrt{300}} = \frac{12(2\sqrt{6})}{3(10\sqrt{3})} = \frac{24\sqrt{6}}{30\sqrt{3}}
\]
### Step 3: Simplify the Coefficient Fraction
Next, simplify the coefficients:
\[
\frac{24}{30} = \frac{4}{5}
\]
So we have:
\[
\frac{4\sqrt{6}}{5\sqrt{3}}
\]
### Step 4: Rationalize the Denominator
Now we need to rationalize:
\[
\frac{4\sqrt{6}}{5\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{4\sqrt{6} \cdot \sqrt{3}}{5 \cdot 3} = \frac{4\sqrt{18}}{15}
\]
### Step 5: Simplify \(\sqrt{18}\)
Now simplify:
\[
\sqrt{18} = \sqrt{9 \cdot 2} = 3\sqrt{2}
\]
### Step 6: Substitute Back
Now substitute it back:
\[
\frac{4(3\sqrt{2})}{15} = \frac{12\sqrt{2}}{15}
\]
### Step 7: Final Simplification
Finally, simplify \(\frac{12}{15}\):
\[
\frac{12}{15} = \frac{4}{5}
\]
So the final expression becomes:
\[
\frac{4\sqrt{2}}{5}
\]
### Conclusion
After carefully going through the calculations again, I confirm that the answer is indeed:
\(\frac{4\sqrt{2}}{5}\).
If there is a specific aspect of this problem that seems incorrect or any specific point of confusion, please let me know and I will be happy to clarify further. Thank you for your patience!