What is the simplest form of [tex]$\frac{2 \sqrt{2}}{\sqrt{3}-\sqrt{2}}$[/tex] ?
A. [tex]$2 \sqrt{6}+4$[/tex]
B. [tex]$2 \sqrt{5}+4$[/tex]
C. [tex]$\frac{2 \sqrt{6}+1}{5}$[/tex]
D. [tex]$\frac{2 \sqrt{5}+4}{5}$[/tex]
Answer (2)
The simplest form of 3 − 2 2 2 is 2 6 + 4 , which corresponds to option A. This was achieved by rationalizing the denominator using its conjugate. The process involves simplifying both the numerator and denominator appropriately to eliminate any square roots in the denominator.
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Multiply the numerator and denominator by the conjugate of the denominator: 3 − 2 2 2 ⋅ 3 + 2 3 + 2 .
Simplify the numerator: 2 2 ( 3 + 2 ) = 2 6 + 4 .
Simplify the denominator: ( 3 − 2 ) ( 3 + 2 ) = 1 .
The simplest form is 2 6 + 4 .
Explanation
Problem Analysis We are given the expression 3 − 2 2 2 and asked to simplify it. Our goal is to rationalize the denominator.
Rationalizing the Denominator To rationalize the denominator, we multiply both the numerator and the denominator by the conjugate of the denominator, which is 3 + 2 . This gives us: 3 − 2 2 2 ⋅ 3 + 2 3 + 2 .
Simplifying the Numerator Now, we multiply out the numerator: 2 2 ( 3 + 2 ) = 2 2 ⋅ 3 + 2 2 ⋅ 2 = 2 6 + 2 ( 2 ) = 2 6 + 4 .
Simplifying the Denominator Next, we multiply out the denominator: ( 3 − 2 ) ( 3 + 2 ) = ( 3 ) 2 − ( 2 ) 2 = 3 − 2 = 1 .
Final Simplification Therefore, the simplified expression is: 1 2 6 + 4 = 2 6 + 4 .
Selecting the Correct Option Comparing our simplified expression 2 6 + 4 with the given options, we see that it matches the first option.
Final Answer Thus, the simplest form of the given expression is 2 6 + 4 .
Examples
Rationalizing the denominator is a useful technique in various fields, such as electrical engineering when dealing with impedance calculations or in physics when simplifying expressions involving wave functions. For example, if you are calculating the total impedance of a circuit and end up with an expression like 3 − 2 5 , you would rationalize the denominator to obtain a more usable form, 5 ( 3 + 2 ) , which makes further calculations easier.
