Question
Simplify the expression \( 15 \sqrt{5}+6 \sqrt{20} \)
Ask by Johnston Schultz. in the United States
Feb 02,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\( 27 \sqrt{5} \)
Solution
To simplify the expression \( 15 \sqrt{5} + 6 \sqrt{20} \), follow these steps:
1. **Simplify \(\sqrt{20}\):**
Break down 20 into its prime factors to simplify the square root.
\[
\sqrt{20} = \sqrt{4 \times 5} = \sqrt{4} \times \sqrt{5} = 2\sqrt{5}
\]
2. **Substitute back into the original expression:**
Replace \(\sqrt{20}\) with \(2\sqrt{5}\).
\[
15 \sqrt{5} + 6 \sqrt{20} = 15 \sqrt{5} + 6 \times 2\sqrt{5}
\]
\[
= 15 \sqrt{5} + 12 \sqrt{5}
\]
3. **Combine like terms:**
Since both terms have \(\sqrt{5}\), you can add their coefficients.
\[
15 \sqrt{5} + 12 \sqrt{5} = (15 + 12) \sqrt{5} = 27 \sqrt{5}
\]
**Final Simplified Expression:**
\[
27 \sqrt{5}
\]
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Simplify this solution The Deep Dive
To simplify the expression \( 15 \sqrt{5}+6 \sqrt{20} \), first simplify \( \sqrt{20} \). We can write \( \sqrt{20} = \sqrt{4 \cdot 5} = \sqrt{4} \cdot \sqrt{5} = 2\sqrt{5} \). Now we can substitute that back into the expression: \[ 15 \sqrt{5} + 6 \sqrt{20} = 15 \sqrt{5} + 6(2\sqrt{5}) = 15 \sqrt{5} + 12 \sqrt{5} \] Now combine the like terms: \[ 15 \sqrt{5} + 12 \sqrt{5} = (15 + 12) \sqrt{5} = 27 \sqrt{5} \] So, the simplified expression is: \[ 27 \sqrt{5} \]
