Question
What is the quotient of the following expression? State the answer in simplest radical form. Rationalize the denominator if necessary. 405√63√ (1 point) Responses 35√7√ the fraction with numerator 3 square root of 5 and denominator square root of 7 35√15 the fraction with numerator square root of 35 and denominator 15 335√7 the fraction with numerator 3 square root of 35 and denominator 7 95√37√
Answers
GPT-4o mini
Let's simplify the expression \( \frac{405 \sqrt{63}}{\sqrt{1}} \).
1. **Simplifying \( \sqrt{63} \)**:
\[
\sqrt{63} = \sqrt{9 \cdot 7} = \sqrt{9} \cdot \sqrt{7} = 3\sqrt{7}
\]
2. **Substituting back**:
Now, substitute \( \sqrt{63} \) into the original expression:
\[
405 \sqrt{63} = 405 \cdot 3\sqrt{7} = 1215\sqrt{7}
\]
3. **Finding the quotient**:
Since the denominator is \( \sqrt{1} = 1 \), the expression simplifies to:
\[
\frac{1215\sqrt{7}}{1} = 1215\sqrt{7}
\]
4. **Simplifying 1215**:
We can also factor 1215:
\[
1215 = 3^5 \times 5 \text{ (since } 1215 = 27 \times 45 = 27 \times 9 \times 5 = 3^3 \times (3^2 \times 5) \text{)}
\]
This simplifies to:
\[
1215 = 15 \cdot 81 = 15 \cdot 9 \cdot 3
\]
In simplest radical terms,
\[
\frac{1215 \sqrt{7}}{1}
\]
Thus, the answer should therefore simplify or present as needed.
Now, looking through the response choices provided, it seems none match exactly.
However:
- **From the options provided, it seems that the best comparison with \( 1215\sqrt{7} \) might be \( 35\sqrt{15} \) but does not match with what we calculated**.
Since there may be an oversight with respecting the available choices to select from, let's provide the key correct simplification:
- It appears we should have \( 1215\sqrt{7} \) as a rational quotient with radical simplified.
I might miss parsing your problem statement accurately on logical framing of quotient or presentation.
Would you want me to elaborate further with another specific breakdown?
1. **Simplifying \( \sqrt{63} \)**:
\[
\sqrt{63} = \sqrt{9 \cdot 7} = \sqrt{9} \cdot \sqrt{7} = 3\sqrt{7}
\]
2. **Substituting back**:
Now, substitute \( \sqrt{63} \) into the original expression:
\[
405 \sqrt{63} = 405 \cdot 3\sqrt{7} = 1215\sqrt{7}
\]
3. **Finding the quotient**:
Since the denominator is \( \sqrt{1} = 1 \), the expression simplifies to:
\[
\frac{1215\sqrt{7}}{1} = 1215\sqrt{7}
\]
4. **Simplifying 1215**:
We can also factor 1215:
\[
1215 = 3^5 \times 5 \text{ (since } 1215 = 27 \times 45 = 27 \times 9 \times 5 = 3^3 \times (3^2 \times 5) \text{)}
\]
This simplifies to:
\[
1215 = 15 \cdot 81 = 15 \cdot 9 \cdot 3
\]
In simplest radical terms,
\[
\frac{1215 \sqrt{7}}{1}
\]
Thus, the answer should therefore simplify or present as needed.
Now, looking through the response choices provided, it seems none match exactly.
However:
- **From the options provided, it seems that the best comparison with \( 1215\sqrt{7} \) might be \( 35\sqrt{15} \) but does not match with what we calculated**.
Since there may be an oversight with respecting the available choices to select from, let's provide the key correct simplification:
- It appears we should have \( 1215\sqrt{7} \) as a rational quotient with radical simplified.
I might miss parsing your problem statement accurately on logical framing of quotient or presentation.
Would you want me to elaborate further with another specific breakdown?