Question
What is the quotient of the following expression? State the answer in the simplest radical form. Rationalize the denominator if necessary.
(8sqrt27)/(2sqrt45)
__sqrt__
(8sqrt27)/(2sqrt45)
__sqrt__
Answers
Answer
try again
it has to be __sqrt__
it has to be __sqrt__
Answer
it means #sqrt# with # being the integers
Answered by
GPT-4o mini
To simplify the expression \(\frac{8\sqrt{27}}{2\sqrt{45}}\), we begin by simplifying the numerator and the denominator.
1. **Simplify the square roots**:
- \( \sqrt{27} = \sqrt{9 \cdot 3} = 3\sqrt{3} \)
- \( \sqrt{45} = \sqrt{9 \cdot 5} = 3\sqrt{5} \)
2. **Substituting back into the expression**:
\[
\frac{8\sqrt{27}}{2\sqrt{45}} = \frac{8(3\sqrt{3})}{2(3\sqrt{5})}
\]
3. **Simplifying the expression**:
- The \(3\) in the numerator and denominator cancels out:
\[
= \frac{8 \cdot 3\sqrt{3}}{2 \cdot 3\sqrt{5}} = \frac{8\sqrt{3}}{2\sqrt{5}}
\]
4. **Further simplification**:
- Now, divide \(8\) by \(2\):
\[
= \frac{8}{2} \cdot \frac{\sqrt{3}}{\sqrt{5}} = 4 \cdot \frac{\sqrt{3}}{\sqrt{5}} = \frac{4\sqrt{3}}{\sqrt{5}}
\]
5. **Rationalize the denominator**:
- To rationalize \(\frac{4\sqrt{3}}{\sqrt{5}}\), multiply the numerator and the denominator by \(\sqrt{5}\):
\[
= \frac{4\sqrt{3} \cdot \sqrt{5}}{\sqrt{5} \cdot \sqrt{5}} = \frac{4\sqrt{15}}{5}
\]
Thus, the answer in the simplest radical form is \(\frac{4\sqrt{15}}{5}\).
1. **Simplify the square roots**:
- \( \sqrt{27} = \sqrt{9 \cdot 3} = 3\sqrt{3} \)
- \( \sqrt{45} = \sqrt{9 \cdot 5} = 3\sqrt{5} \)
2. **Substituting back into the expression**:
\[
\frac{8\sqrt{27}}{2\sqrt{45}} = \frac{8(3\sqrt{3})}{2(3\sqrt{5})}
\]
3. **Simplifying the expression**:
- The \(3\) in the numerator and denominator cancels out:
\[
= \frac{8 \cdot 3\sqrt{3}}{2 \cdot 3\sqrt{5}} = \frac{8\sqrt{3}}{2\sqrt{5}}
\]
4. **Further simplification**:
- Now, divide \(8\) by \(2\):
\[
= \frac{8}{2} \cdot \frac{\sqrt{3}}{\sqrt{5}} = 4 \cdot \frac{\sqrt{3}}{\sqrt{5}} = \frac{4\sqrt{3}}{\sqrt{5}}
\]
5. **Rationalize the denominator**:
- To rationalize \(\frac{4\sqrt{3}}{\sqrt{5}}\), multiply the numerator and the denominator by \(\sqrt{5}\):
\[
= \frac{4\sqrt{3} \cdot \sqrt{5}}{\sqrt{5} \cdot \sqrt{5}} = \frac{4\sqrt{15}}{5}
\]
Thus, the answer in the simplest radical form is \(\frac{4\sqrt{15}}{5}\).
Answered by
GPT-4o mini
I apologize for the confusion. Let's redo the calculation step-by-step to ensure the result is presented as requested.
We start with the expression:
\[
\frac{8\sqrt{27}}{2\sqrt{45}}
\]
1. **Simplifying \(\sqrt{27}\) and \(\sqrt{45}\)**:
- \(\sqrt{27} = \sqrt{9 \cdot 3} = 3\sqrt{3}\)
- \(\sqrt{45} = \sqrt{9 \cdot 5} = 3\sqrt{5}\)
2. **Substituting these back into the expression**:
\[
\frac{8\sqrt{27}}{2\sqrt{45}} = \frac{8(3\sqrt{3})}{2(3\sqrt{5})}
\]
3. **Cancel the common terms**:
- The \(3\) in the numerator and denominator cancel out:
\[
= \frac{8 \cdot \sqrt{3}}{2 \cdot \sqrt{5}} = \frac{8}{2} \cdot \frac{\sqrt{3}}{\sqrt{5}} = 4 \cdot \frac{\sqrt{3}}{\sqrt{5}} = \frac{4\sqrt{3}}{\sqrt{5}}
\]
4. **Rationalizing the denominator** (to format it \(\_\sqrt{}\)):
- Multiply the numerator and the denominator by \(\sqrt{5}\):
\[
\frac{4\sqrt{3}}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}} = \frac{4\sqrt{15}}{5}
\]
Now, expressing this in the requested \(\_\sqrt{}\) format, we write:
\[
\text{Final Result: } \frac{4\sqrt{15}}{5} \quad \text{(no additional simplification in sqrt form)}
\]
I realize you intended for the final answer to be explicitly in the radical form designated (\_\sqrt{}).
However, if you're specifically asking about a radical expression itself, then it seems you want something like:
\[
\sqrt{A}, \text{ where } A = B \text{/ some integer}
\]
It appears there might have been some misinterpretation. The direct quotient \(\frac{4\sqrt{15}}{5}\) remains the most simplified radical form, but does not fit into the \_\sqrt{} format.
If you could clarify or provide the context for \(\_\sqrt{}\), it would help tailor the answer more appropriately! Thank you!
We start with the expression:
\[
\frac{8\sqrt{27}}{2\sqrt{45}}
\]
1. **Simplifying \(\sqrt{27}\) and \(\sqrt{45}\)**:
- \(\sqrt{27} = \sqrt{9 \cdot 3} = 3\sqrt{3}\)
- \(\sqrt{45} = \sqrt{9 \cdot 5} = 3\sqrt{5}\)
2. **Substituting these back into the expression**:
\[
\frac{8\sqrt{27}}{2\sqrt{45}} = \frac{8(3\sqrt{3})}{2(3\sqrt{5})}
\]
3. **Cancel the common terms**:
- The \(3\) in the numerator and denominator cancel out:
\[
= \frac{8 \cdot \sqrt{3}}{2 \cdot \sqrt{5}} = \frac{8}{2} \cdot \frac{\sqrt{3}}{\sqrt{5}} = 4 \cdot \frac{\sqrt{3}}{\sqrt{5}} = \frac{4\sqrt{3}}{\sqrt{5}}
\]
4. **Rationalizing the denominator** (to format it \(\_\sqrt{}\)):
- Multiply the numerator and the denominator by \(\sqrt{5}\):
\[
\frac{4\sqrt{3}}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}} = \frac{4\sqrt{15}}{5}
\]
Now, expressing this in the requested \(\_\sqrt{}\) format, we write:
\[
\text{Final Result: } \frac{4\sqrt{15}}{5} \quad \text{(no additional simplification in sqrt form)}
\]
I realize you intended for the final answer to be explicitly in the radical form designated (\_\sqrt{}).
However, if you're specifically asking about a radical expression itself, then it seems you want something like:
\[
\sqrt{A}, \text{ where } A = B \text{/ some integer}
\]
It appears there might have been some misinterpretation. The direct quotient \(\frac{4\sqrt{15}}{5}\) remains the most simplified radical form, but does not fit into the \_\sqrt{} format.
If you could clarify or provide the context for \(\_\sqrt{}\), it would help tailor the answer more appropriately! Thank you!
Answered by
GPT-4o mini
Thank you for clarifying the request! Let's express the quotient in the format you specified, using "#" to denote integers in the radical.
Given the expression:
\[
\frac{8\sqrt{27}}{2\sqrt{45}}
\]
1. **Simplifying the square roots**:
- \(\sqrt{27} = 3\sqrt{3}\)
- \(\sqrt{45} = 3\sqrt{5}\)
2. **Substituting back into the expression**:
\[
\frac{8\sqrt{27}}{2\sqrt{45}} = \frac{8(3\sqrt{3})}{2(3\sqrt{5})}
\]
3. **Canceling the common factors**:
- The \(3\) cancels out, and we simplify further:
\[
= \frac{8\sqrt{3}}{2\sqrt{5}} = \frac{8}{2} \cdot \frac{\sqrt{3}}{\sqrt{5}} = 4 \cdot \frac{\sqrt{3}}{\sqrt{5}} = \frac{4\sqrt{3}}{\sqrt{5}}
\]
4. **Rationalizing the denominator**:
- Multiply the numerator and the denominator by \(\sqrt{5}\):
\[
= \frac{4\sqrt{3} \cdot \sqrt{5}}{\sqrt{5} \cdot \sqrt{5}} = \frac{4\sqrt{15}}{5}
\]
Now, to express this final result according to your specified format \#\sqrt\#:
\[
= \frac{4\sqrt{15}}{5} = 4\sqrt{15} \div 5
\]
In a way to denote this as \(\#\sqrt{}\):
**Final Answer:**
\[
\boxed{\frac{4 \sqrt{15}}{5}} \text{ or express it as } \frac{4}{5} \sqrt{15}
\]
Please let me know if you need any other adjustments!
Given the expression:
\[
\frac{8\sqrt{27}}{2\sqrt{45}}
\]
1. **Simplifying the square roots**:
- \(\sqrt{27} = 3\sqrt{3}\)
- \(\sqrt{45} = 3\sqrt{5}\)
2. **Substituting back into the expression**:
\[
\frac{8\sqrt{27}}{2\sqrt{45}} = \frac{8(3\sqrt{3})}{2(3\sqrt{5})}
\]
3. **Canceling the common factors**:
- The \(3\) cancels out, and we simplify further:
\[
= \frac{8\sqrt{3}}{2\sqrt{5}} = \frac{8}{2} \cdot \frac{\sqrt{3}}{\sqrt{5}} = 4 \cdot \frac{\sqrt{3}}{\sqrt{5}} = \frac{4\sqrt{3}}{\sqrt{5}}
\]
4. **Rationalizing the denominator**:
- Multiply the numerator and the denominator by \(\sqrt{5}\):
\[
= \frac{4\sqrt{3} \cdot \sqrt{5}}{\sqrt{5} \cdot \sqrt{5}} = \frac{4\sqrt{15}}{5}
\]
Now, to express this final result according to your specified format \#\sqrt\#:
\[
= \frac{4\sqrt{15}}{5} = 4\sqrt{15} \div 5
\]
In a way to denote this as \(\#\sqrt{}\):
**Final Answer:**
\[
\boxed{\frac{4 \sqrt{15}}{5}} \text{ or express it as } \frac{4}{5} \sqrt{15}
\]
Please let me know if you need any other adjustments!
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