Question
Simplify. Assume \( f \) is greater than or equal to zero. \( \sqrt{18 f^{8}} \)
Ask by Bond Reed. in the United States
Jan 24,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The simplified form of \(\sqrt{18 f^{8}}\) is \(3 f^{4} \sqrt{2}\).
Solution
To simplify the expression \(\sqrt{18 f^{8}}\) with the condition that \(f \geq 0\), follow these steps:
1. **Factor the Radicand:**
\[
\sqrt{18 f^{8}} = \sqrt{9 \times 2 \times f^{8}}
\]
2. **Simplify the Square Root:**
- \(\sqrt{9} = 3\)
- \(\sqrt{f^{8}} = f^{4}\) (since \(f \geq 0\))
- \(\sqrt{2}\) remains as it is because 2 is not a perfect square.
3. **Combine the Simplified Parts:**
\[
\sqrt{18 f^{8}} = 3 f^{4} \sqrt{2}
\]
**Final Simplified Form:**
\[
3 f^{4} \sqrt{2}
\]
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Simplify this solution Extra Insights
To simplify \( \sqrt{18 f^{8}} \), we can break it down. First, we recognize that \( 18 \) can be rewritten as \( 9 \times 2 \), and \( f^{8} \) is a perfect square. Thus: \[ \sqrt{18 f^{8}} = \sqrt{9 \times 2 \times f^{8}}. \] Now we can simplify each part separately: \[ \sqrt{9} = 3, \] \[ \sqrt{f^{8}} = f^{4}. \] Putting it all together, we have: \[ \sqrt{18 f^{8}} = 3f^{4} \sqrt{2}. \] So, the simplified result is: \[ 3f^{4} \sqrt{2}. \]
