Multiply $\frac{3}{\sqrt{17}-\sqrt{2}}$ by which fraction will produce an equivalent fraction with a rational denominator?
A. $\frac{\sqrt{17}-\sqrt{2}}{\sqrt{17}-\sqrt{2}}$
B. $\frac{\sqrt{17}+\sqrt{2}}{\sqrt{17}+\sqrt{2}}$
C. $\frac{\sqrt{2}-\sqrt{17}}{\sqrt{2}-\sqrt{17}}$
D. $\frac{\sqrt{15}}{\sqrt{15}}$
Answer (2)
To rationalize the denominator of the fraction 17 − 2 3 , we multiply it by 17 + 2 17 + 2 , which is the conjugate of the denominator. This results in a rational denominator of 15. The selected option is B .
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Identify the conjugate of the denominator: The conjugate of 17 − 2 is 17 + 2 .
Multiply the given fraction by a fraction equal to 1, using the conjugate in both the numerator and denominator: 17 + 2 17 + 2 .
Simplify the new denominator using the difference of squares: ( 17 − 2 ) ( 17 + 2 ) = 17 − 2 = 15 .
The fraction that produces an equivalent fraction with a rational denominator is 17 + 2 17 + 2 .
Explanation
Understanding the Problem We are given the fraction 17 − 2 3 and asked to find a fraction to multiply it by such that the resulting fraction has a rational denominator. This process is called rationalizing the denominator.
Finding the Conjugate To rationalize the denominator, we need to multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of 17 − 2 is 17 + 2 .
Multiplying by the Conjugate Therefore, we need to multiply the given fraction by 17 + 2 17 + 2 . Let's perform the multiplication: 17 − 2 3 × 17 + 2 17 + 2 = ( 17 − 2 ) ( 17 + 2 ) 3 ( 17 + 2 ) .
Simplifying the Denominator Now, let's simplify the denominator using the difference of squares formula, ( a − b ) ( a + b ) = a 2 − b 2 : ( 17 − 2 ) ( 17 + 2 ) = ( 17 ) 2 − ( 2 ) 2 = 17 − 2 = 15. So the denominator becomes 15, which is a rational number.
Conclusion Thus, multiplying by 17 + 2 17 + 2 rationalizes the denominator.
Examples
Rationalizing the denominator is a useful technique in various fields, such as physics and engineering, where simplified expressions are easier to work with. For example, when calculating impedance in electrical circuits or dealing with wave functions in quantum mechanics, rationalizing denominators can make calculations more manageable and lead to clearer insights into the problem.
