Rationalize the denominator: [tex]\frac{2}{3 \sqrt{2}-5}[/tex]
Answer (2)
Multiply the numerator and denominator by the conjugate of the denominator: 3 2 − 5 2 × 3 2 + 5 3 2 + 5 .
Simplify the denominator using the difference of squares: ( 3 2 − 5 ) ( 3 2 + 5 ) = − 7 .
Distribute the 2 in the numerator: − 7 2 ( 3 2 + 5 ) = − 7 6 2 + 10 .
Rewrite the expression: − 7 10 + 6 2 . The final answer is − 7 10 + 6 2 .
Explanation
Understanding the Problem We are asked to rationalize the denominator of the fraction 3 2 − 5 2 . Rationalizing the denominator means rewriting the fraction so that there are no radical expressions in 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 − 5 is 3 2 + 5 .
Multiplying by the Conjugate Multiply the numerator and denominator by the conjugate: 3 2 − 5 2 × 3 2 + 5 3 2 + 5 = ( 3 2 − 5 ) ( 3 2 + 5 ) 2 ( 3 2 + 5 )
Simplifying the Denominator Now, we expand the denominator using the difference of squares formula, ( a − b ) ( a + b ) = a 2 − b 2 , where a = 3 2 and b = 5 :
( 3 2 − 5 ) ( 3 2 + 5 ) = ( 3 2 ) 2 − 5 2 = 9 ( 2 ) − 25 = 18 − 25 = − 7
Substituting the Simplified Denominator So, the expression becomes: − 7 2 ( 3 2 + 5 )
Distributing in the Numerator Distribute the 2 in the numerator: − 7 6 2 + 10
Final Answer Finally, we can rewrite the expression as: − 7 6 2 + 10 or − 7 10 + 6 2
Conclusion Thus, the rationalized form of the given expression is − 7 10 + 6 2 .
Examples
Rationalizing the denominator is a useful technique in various mathematical contexts. For instance, when dealing with electrical circuits, you might encounter expressions with radicals in the denominator when calculating impedance. Rationalizing the denominator simplifies further calculations and makes it easier to compare different circuit configurations. Similarly, in optics, when calculating refractive indices, rationalizing denominators can help in simplifying expressions and making them easier to interpret.
To rationalize the denominator of 3 2 − 5 2 , multiply the numerator and denominator by the conjugate 3 2 + 5 . This simplifies to − 7 10 + 6 2 , removing the radical from the denominator. Rationalizing helps in simplifying expressions for easier calculation and interpretation.
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