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Use the product rule to simplify the radical. \[ \sqrt{45} \]

Ask by Donnelly Ellis. in the United States
Jan 22,2025

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\[ \sqrt{45} = 3\sqrt{5} \]

Solución

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: \[ 3\sqrt{5} \]

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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?

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