Question
Multiply, and then simplify if possible. Assume that all variables represent positive real numbers. \( \sqrt{2 x}(\sqrt{2}-\sqrt{x}) \) \( \sqrt{2 x}(\sqrt{2}-\sqrt{x})=\square \) (Simplify your answer. Type an exact answer, using radicals as needed.)
Ask by Matthews Campos. in the United States
Mar 14,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( \sqrt{2x}(\sqrt{2}-\sqrt{x}) = 2\sqrt{x} - \sqrt{2}x \)
Solution
Calculate or simplify the expression \( \sqrt{2x}(\sqrt{2}-\sqrt{x}) \).
Simplify the expression by following steps:
- step0: Solution:
\(\sqrt{2x}\times \left(\sqrt{2}-\sqrt{x}\right)\)
- step1: Distribute \(\sqrt{2x}:\)
\(\sqrt{2x}\times \sqrt{2}+\sqrt{2x}\times \left(-\sqrt{x}\right)\)
- step2: Calculate the product:
\(2\sqrt{x}-\sqrt{2}\times x\)
The simplified form of \( \sqrt{2x}(\sqrt{2}-\sqrt{x}) \) is \( 2\sqrt{x}-\sqrt{2}x \).
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Simplify this solution The Deep Dive
To multiply \( \sqrt{2x}(\sqrt{2} - \sqrt{x}) \), we can use the distributive property (also known as the FOIL method for binomials): 1. Multiply \( \sqrt{2x} \) by \( \sqrt{2} \): \[ \sqrt{2x} \cdot \sqrt{2} = \sqrt{4x} = 2\sqrt{x} \] 2. Multiply \( \sqrt{2x} \) by \( -\sqrt{x} \): \[ \sqrt{2x} \cdot -\sqrt{x} = -\sqrt{2x^2} = -\sqrt{2}x \] Now, combine the two results: \[ \sqrt{2x}(\sqrt{2} - \sqrt{x}) = 2\sqrt{x} - \sqrt{2}x \] So, the simplified expression is: \[ 2\sqrt{x} - \sqrt{2}x \]
