Question
What is the simplest form of \( \frac{2 \sqrt{2}}{\sqrt{3}-\sqrt{2}} \) ?
Ask by Santiago Carlson. in the United States
Mar 11,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The simplest form is \( 2\sqrt{6}+4 \).
Solution
Calculate or simplify the expression \( \frac{2 \sqrt{2}}{\sqrt{3}-\sqrt{2}} \).
Calculate the value by following steps:
- step0: Calculate:
\(\frac{2\sqrt{2}}{\sqrt{3}-\sqrt{2}}\)
- step1: Multiply by the Conjugate:
\(\frac{2\sqrt{2}\times \left(\sqrt{3}+\sqrt{2}\right)}{\left(\sqrt{3}-\sqrt{2}\right)\left(\sqrt{3}+\sqrt{2}\right)}\)
- step2: Multiply the numbers:
\(\frac{2\sqrt{2}\times \left(\sqrt{3}+\sqrt{2}\right)}{1}\)
- step3: Divide the numbers:
\(2\sqrt{2}\times \left(\sqrt{3}+\sqrt{2}\right)\)
- step4: Multiply the terms:
\(\left(2\sqrt{3}+2\sqrt{2}\right)\sqrt{2}\)
- step5: Apply the distributive property:
\(2\sqrt{3}\times \sqrt{2}+2\sqrt{2}\times \sqrt{2}\)
- step6: Multiply the terms:
\(2\sqrt{6}+2\sqrt{2}\times \sqrt{2}\)
- step7: Multiply the numbers:
\(2\sqrt{6}+4\)
The simplest form of \( \frac{2 \sqrt{2}}{\sqrt{3}-\sqrt{2}} \) is \( 2\sqrt{6}+4 \).
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Simplify this solution Mind Expander
To simplify \( \frac{2 \sqrt{2}}{\sqrt{3}-\sqrt{2}} \), we can rationalize the denominator by multiplying the numerator and the denominator by the conjugate of the denominator, \( \sqrt{3} + \sqrt{2} \): \[ \frac{2 \sqrt{2} (\sqrt{3} + \sqrt{2})}{(\sqrt{3} - \sqrt{2})(\sqrt{3} + \sqrt{2})} \] In the denominator, \( (\sqrt{3} - \sqrt{2})(\sqrt{3} + \sqrt{2}) \) simplifies to \( 3 - 2 = 1 \). Thus, we have: \[ 2 \sqrt{2} (\sqrt{3} + \sqrt{2}) \] Now, distributing \( 2 \sqrt{2} \): \[ = 2 \sqrt{6} + 4 \] So the simplified form is: \[ \boxed{2\sqrt{6} + 4} \]
