Simplify $\frac{7}{3-\sqrt{2}}$
Answer (2)
The expression 3 − 2 7 can be simplified by multiplying by its conjugate to get 3 + 2 after rationalizing the denominator. The complete steps involve expanding and simplifying the fraction. The final answer is 3 + 2 .
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Multiply the numerator and denominator by the conjugate of the denominator: 3 − 2 7 × 3 + 2 3 + 2 .
Expand the denominator using the difference of squares: ( 3 ) 2 − ( 2 ) 2 7 ( 3 + 2 ) = 9 − 2 7 ( 3 + 2 ) .
Simplify the denominator: 7 7 ( 3 + 2 ) .
Cancel the common factor of 7: 3 + 2 . The simplified expression is 3 + 2 .
Explanation
Understanding the Problem We are asked to simplify the expression 3 − 2 7 . This involves rationalizing the denominator.
Finding the Conjugate To rationalize the denominator, we multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of 3 − 2 is 3 + 2 .
Multiplying by the Conjugate Multiply the numerator and denominator by the conjugate: 3 − 2 7 × 3 + 2 3 + 2 = ( 3 − 2 ) ( 3 + 2 ) 7 ( 3 + 2 )
Expanding the Denominator Expand the denominator using the difference of squares formula, ( a − b ) ( a + b ) = a 2 − b 2 : ( 3 ) 2 − ( 2 ) 2 7 ( 3 + 2 ) = 9 − 2 7 ( 3 + 2 ) = 7 7 ( 3 + 2 )
Simplifying the Expression Cancel the common factor of 7 in the numerator and the denominator: 7 7 ( 3 + 2 ) = 3 + 2
Final Answer Therefore, the simplified expression is 3 + 2 .
Examples
Rationalizing the denominator is a technique used in various fields, such as physics and engineering, when dealing with complex numbers or expressions in calculations. For example, when calculating impedance in electrical circuits or dealing with wave functions in quantum mechanics, it's often necessary to rationalize denominators to simplify expressions and obtain more manageable results. This technique ensures that the final result is expressed in a standard form, making it easier to interpret and use in further calculations. Suppose you are designing an electronic filter and one of your calculations results in a transfer function with a term like 1 − 3 1 . To make this more practical for implementation, you would rationalize the denominator to get − 2 1 + 3 , which is easier to work with.
