Question
Use the product rule to simplify the radical. \[ \sqrt{45} \]
Ask by Donnelly Ellis. in the United States
Jan 22,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\[
\sqrt{45} = 3\sqrt{5}
\]
Solution
To simplify the radical \(\sqrt{45}\) using the **product rule**, follow these steps:
1. **Factor the number inside the radical into its prime factors:**
\(45 = 9 \times 5\)
Notice that \(9\) is a perfect square (\(9 = 3^2\)).
2. **Apply the product rule of radicals:**
\[
\sqrt{45} = \sqrt{9 \times 5} = \sqrt{9} \times \sqrt{5}
\]
3. **Simplify the square root of the perfect square:**
\[
\sqrt{9} = 3
\]
4. **Combine the simplified terms:**
\[
3 \times \sqrt{5} = 3\sqrt{5}
\]
So, the simplified form of \(\sqrt{45}\) is:
\[
3\sqrt{5}
\]
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Beyond the Answer
To simplify \( \sqrt{45} \), we can break it down into its factors. Notice that \( 45 = 9 \times 5 \). We know that \( \sqrt{9} = 3 \), so we can rewrite \( \sqrt{45} \) as \( \sqrt{9 \times 5} = \sqrt{9} \times \sqrt{5} = 3\sqrt{5} \). Thus, \( \sqrt{45} = 3\sqrt{5} \). Now you've got a simplified expression! Isn’t it satisfying to unravel a radical like this?
