Question
Simplify. Assume \( d \) is greater than or equal to zero. \[ \sqrt{45 d^{10}} \]
Ask by Jimenez Rodriquez. in the United States
Jan 24,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\(3d^{5}\sqrt{5}\)
Solution
To simplify the expression \(\sqrt{45 d^{10}}\) with the assumption that \(d \geq 0\), follow these steps:
1. **Separate the Square Root:**
\[
\sqrt{45 d^{10}} = \sqrt{45} \cdot \sqrt{d^{10}}
\]
2. **Simplify \(\sqrt{d^{10}}\):**
Since \(d \geq 0\),
\[
\sqrt{d^{10}} = d^{5}
\]
3. **Simplify \(\sqrt{45}\):**
Break down 45 into its prime factors:
\[
45 = 9 \times 5 = 3^2 \times 5
\]
Therefore,
\[
\sqrt{45} = \sqrt{3^2 \times 5} = 3\sqrt{5}
\]
4. **Combine the Simplified Parts:**
\[
\sqrt{45 d^{10}} = 3\sqrt{5} \cdot d^5 = 3d^5\sqrt{5}
\]
**Final Simplified Form:**
\[
3d^{5} \sqrt{5}
\]
Answer: \(3\, d^{5} \sqrt{5}\)
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Simplify this solution Mind Expander
To simplify \( \sqrt{45 d^{10}} \), we can break it down into its components. Start with the number inside the square root: \[ \sqrt{45 d^{10}} = \sqrt{45} \cdot \sqrt{d^{10}} \] First, simplify \( \sqrt{45} \). We can factor it as follows: \[ 45 = 9 \cdot 5 = 3^2 \cdot 5 \] So, \[ \sqrt{45} = \sqrt{9 \cdot 5} = \sqrt{9} \cdot \sqrt{5} = 3\sqrt{5} \] Next, simplify \( \sqrt{d^{10}} \): \[ \sqrt{d^{10}} = d^{10/2} = d^5 \] Now, combine both simplified parts: \[ \sqrt{45 d^{10}} = 3\sqrt{5} \cdot d^5 \] Thus, the final simplified expression is: \[ 3d^5\sqrt{5} \]
