Rationalize the denominator for $\frac{15}{\sqrt{7}+\sqrt{2}}$ and simplify to the simplest form.
Answer (1)
Multiply the numerator and denominator by the conjugate of the denominator: 7 + 2 15 × 7 − 2 7 − 2 .
Simplify the denominator using the difference of squares: ( 7 + 2 ) ( 7 − 2 ) = 7 − 2 = 5 .
Simplify the fraction: 5 15 ( 7 − 2 ) = 3 ( 7 − 2 ) .
Distribute to obtain the final simplified expression: 3 7 − 3 2 .
Explanation
Understanding the Problem We are given the expression 7 + 2 15 and asked to rationalize the denominator and simplify. Rationalizing the denominator means eliminating any radical expressions from the denominator.
Multiplying by the Conjugate To rationalize the denominator, we multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of 7 + 2 is 7 − 2 . So, we multiply the given expression by 7 − 2 7 − 2 : 7 + 2 15 × 7 − 2 7 − 2 = ( 7 + 2 ) ( 7 − 2 ) 15 ( 7 − 2 ) .
Simplifying the Denominator Now, we simplify the denominator using the difference of squares formula, ( a + b ) ( a − b ) = a 2 − b 2 : ( 7 + 2 ) ( 7 − 2 ) = ( 7 ) 2 − ( 2 ) 2 = 7 − 2 = 5. So the expression becomes: 5 15 ( 7 − 2 ) .
Simplifying the Fraction Next, we simplify the fraction by dividing 15 by 5: 5 15 ( 7 − 2 ) = 3 ( 7 − 2 ) .
Distributing and Final Answer Finally, we distribute the 3 to get the simplified expression: 3 ( 7 − 2 ) = 3 7 − 3 2 .
Examples
Rationalizing the denominator is a technique used in various fields, such as physics and engineering, when dealing with equations involving radicals. For instance, when calculating impedance in electrical circuits or dealing with wave functions in quantum mechanics, it's often necessary to simplify expressions with radicals in the denominator to obtain more manageable and interpretable results. This technique ensures that the final result is expressed in a standard form, making it easier to perform further calculations or compare different quantities.
