Question
Multiply the special product. \[ (\sqrt{5 m}+7 \sqrt{n})^{2}=\square \]
Ask by Park Burns. in the United States
Dec 03,2024
Upstudy AI Solution
Tutor-Verified Answer
Answer
\(5m+14\sqrt{5mn}+49n\)
Solution
Calculate or simplify the expression \( (\sqrt{5m}+7\sqrt{n})^{2} \).
Simplify the expression by following steps:
- step0: Solution:
\(\left(\sqrt{5m}+7\sqrt{n}\right)^{2}\)
- step1: Calculate:
\(5m+14\sqrt{5mn}+49n\)
The special product \((\sqrt{5m}+7\sqrt{n})^{2}\) simplifies to \(5m+14\sqrt{5mn}+49n\).
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Simplify this solution The Deep Dive
To multiply the special product \((\sqrt{5m} + 7\sqrt{n})^2\), we apply the formula \((a + b)^2 = a^2 + 2ab + b^2\). Here, \(a = \sqrt{5m}\) and \(b = 7\sqrt{n}\). Calculating each term: 1. \(a^2 = (\sqrt{5m})^2 = 5m\) 2. \(b^2 = (7\sqrt{n})^2 = 49n\) 3. \(2ab = 2(\sqrt{5m})(7\sqrt{n}) = 14\sqrt{5mn}\) Putting it all together: \[ (\sqrt{5m} + 7\sqrt{n})^2 = 5m + 14\sqrt{5mn} + 49n \] Thus, the final result is: \[ 5m + 14\sqrt{5mn} + 49n = \square \]
