\frac{6-4 \sqrt{2}}{6+4 \sqrt{2}}
Answer (2)
Multiply the numerator and denominator by the conjugate of the denominator.
Expand the numerator and the denominator.
Simplify the expression by dividing both terms in the numerator by the denominator.
The simplified expression is 17 − 12 2 .
Explanation
Problem Analysis We are given the expression 6 + 4 2 6 − 4 2 and our goal is to simplify it.
Rationalizing the Denominator To simplify this expression, we will multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of 6 + 4 2 is 6 − 4 2 . This will help us eliminate the square root from the denominator.
Multiplying by the Conjugate Multiply the numerator and denominator by the conjugate: 6 + 4 2 6 − 4 2 ⋅ 6 − 4 2 6 − 4 2
Expanding Numerator and Denominator Expand the numerator: ( 6 − 4 2 ) ( 6 − 4 2 ) = 6 ( 6 ) + 6 ( − 4 2 ) − 4 2 ( 6 ) + ( − 4 2 ) ( − 4 2 ) = 36 − 24 2 − 24 2 + 16 ( 2 ) = 36 − 48 2 + 32 = 68 − 48 2 Expand the denominator: ( 6 + 4 2 ) ( 6 − 4 2 ) = 6 ( 6 ) + 6 ( − 4 2 ) + 4 2 ( 6 ) + ( 4 2 ) ( − 4 2 ) = 36 − 24 2 + 24 2 − 16 ( 2 ) = 36 − 32 = 4
Simplified Expression Now we have: 4 68 − 48 2
Final Simplification Divide both terms in the numerator by 4: 4 68 − 4 48 2 = 17 − 12 2
Final Answer Therefore, the simplified expression is 17 − 12 2 .
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 complex numbers. For example, if you are calculating the impedance of a circuit and end up with a complex number in the denominator, you would rationalize it to make further calculations easier. This technique ensures that the final result is expressed in a standard form, making it easier to interpret and use in subsequent calculations. Simplifying radical expressions helps in obtaining more accurate and manageable results in practical applications.
To simplify the expression 6 + 4 2 6 − 4 2 , we multiply by the conjugate of the denominator, simplify, and find that the result is 17 − 12 2 .
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