Question
Use the product rule to simplify the radical. \( \sqrt{98} \)
Ask by Hills Newton. in the United States
Jan 22,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\( \sqrt{98} = 7\sqrt{2} \)
Solution
To simplify the radical \( \sqrt{98} \) using the product rule, follow these steps:
**Product Rule for Radicals:**
\[
\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}
\]
1. **Factor 98 into a Product of Perfect Squares and Other Factors:**
\( 98 = 49 \times 2 \)
Here, 49 is a perfect square since \( 49 = 7^2 \).
2. **Apply the Product Rule:**
\[
\sqrt{98} = \sqrt{49 \times 2} = \sqrt{49} \times \sqrt{2}
\]
3. **Simplify the Square Roots:**
\[
\sqrt{49} = 7
\]
\[
\sqrt{2} \text{ remains as is since 2 is not a perfect square.}
\]
4. **Combine the Simplified Terms:**
\[
\sqrt{98} = 7 \times \sqrt{2} = 7\sqrt{2}
\]
**Final Simplified Form:**
\[
\sqrt{98} = 7\sqrt{2}
\]
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Simplify this solution Extra Insights
To simplify \( \sqrt{98} \), we start by factoring 98 into its prime factors. We find that \( 98 = 49 \times 2 \), and since \( 49 \) is a perfect square, we can apply the product rule for square roots. Thus, \( \sqrt{98} = \sqrt{49 \times 2} = \sqrt{49} \times \sqrt{2} = 7\sqrt{2} \). Not only does this give a clearer expression, but it also illustrates how utilizing the product rule can make simplifications more straightforward! By recognizing perfect squares, you'll find that simplifying radicals becomes much easier and more efficient. Next time you're faced with a radical, look for ways to break it down using similar techniques for faster results!
