Question
Use the product rule to simplify the radical. \[ \sqrt{117} \]
Ask by Goodwin Fleming. in the United States
Jan 22,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\[
\sqrt{117} = 3\sqrt{13}
\]
Solution
To simplify the radical \(\sqrt{117}\) using the product rule, follow these steps:
**Step 1: Factor the Number Inside the Radical**
Find the prime factors of 117 to identify any perfect squares.
\[
117 = 9 \times 13
\]
Here, 9 is a perfect square because \(9 = 3^2\).
**Step 2: Apply the Product Rule for Radicals**
The product rule states that \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\). Apply this to \(\sqrt{117}\):
\[
\sqrt{117} = \sqrt{9 \times 13} = \sqrt{9} \times \sqrt{13}
\]
**Step 3: Simplify the Square Root of the Perfect Square**
\[
\sqrt{9} = 3
\]
**Step 4: Combine the Results**
\[
\sqrt{117} = 3 \times \sqrt{13} = 3\sqrt{13}
\]
**Final Answer:**
\[
\sqrt{117} = 3\sqrt{13}
\]
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Beyond the Answer
Did you know that the number 117 has a fascinating prime factorization? It can be broken down as \( 3 \times 39 \) and then \( 39 \) can be further factored into \( 3 \times 13 \). This means \( 117 = 3^2 \times 13 \). Using the product rule for square roots, we can simplify \( \sqrt{117} \) into \( \sqrt{3^2 \times 13} \), which equals \( 3\sqrt{13} \). To grasp this concept better, think of square roots as pairs. Every time you find a perfect square within a radical, you can pull it out. With \( \sqrt{117} \), you recognized \( 3^2 \) as the perfect square, allowing you to simplify the expression joyfully, making \( \sqrt{117} \) much easier to work with. Happy simplifying!
