Question
Simplify: \( 5 \sqrt{54}+2 \sqrt{175}-1 \sqrt{24}-5 \sqrt{7} \) (entry tip: to type type "5sqrt(3x)" - no quotes) \( \square> \) Next Question
Ask by Pollard Beck. in the United States
Jan 25,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\( 13 \sqrt{6} + 5 \sqrt{7} \)
Solution
Let's simplify the expression step by step:
\[
5\sqrt{54} + 2\sqrt{175} - \sqrt{24} - 5\sqrt{7}
\]
1. **Simplify each square root:**
- \(\sqrt{54} = \sqrt{9 \times 6} = 3\sqrt{6}\)
- \(\sqrt{175} = \sqrt{25 \times 7} = 5\sqrt{7}\)
- \(\sqrt{24} = \sqrt{4 \times 6} = 2\sqrt{6}\)
2. **Substitute back into the expression:**
\[
5 \times 3\sqrt{6} + 2 \times 5\sqrt{7} - 2\sqrt{6} - 5\sqrt{7}
\]
3. **Multiply the coefficients:**
\[
15\sqrt{6} + 10\sqrt{7} - 2\sqrt{6} - 5\sqrt{7}
\]
4. **Combine like terms:**
- Combine the \(\sqrt{6}\) terms: \(15\sqrt{6} - 2\sqrt{6} = 13\sqrt{6}\)
- Combine the \(\sqrt{7}\) terms: \(10\sqrt{7} - 5\sqrt{7} = 5\sqrt{7}\)
5. **Final simplified expression:**
\[
13\sqrt{6} + 5\sqrt{7}
\]
**Answer:**
\[
13 \sqrt{6} + 5 \sqrt{7}
\]
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Simplify this solution Extra Insights
To simplify \( 5 \sqrt{54}+2 \sqrt{175}-1 \sqrt{24}-5 \sqrt{7} \), we start by simplifying each square root: 1. \( \sqrt{54} = \sqrt{9 \cdot 6} = 3\sqrt{6} \) so \( 5\sqrt{54} = 15\sqrt{6} \). 2. \( \sqrt{175} = \sqrt{25 \cdot 7} = 5\sqrt{7} \) so \( 2\sqrt{175} = 10\sqrt{7} \). 3. \( \sqrt{24} = \sqrt{4 \cdot 6} = 2\sqrt{6} \) so \( -1\sqrt{24} = -2\sqrt{6} \). 4. \( -5\sqrt{7} \) stays as is. Now combine all the terms: - Collect \( \sqrt{6} \) terms: \( 15\sqrt{6} - 2\sqrt{6} = 13\sqrt{6} \). - Collect \( \sqrt{7} \) terms: \( 10\sqrt{7} - 5\sqrt{7} = 5\sqrt{7} \). Thus, the simplified expression is: \( \boxed{13\sqrt{6} + 5\sqrt{7}} \)
