Multiplying $\frac{3}{\sqrt{17}-\sqrt{2}}$ 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)
Identify the conjugate of the denominator: The conjugate of 17 − 2 is 17 + 2 .
Multiply the fraction by the conjugate over itself: 17 − 2 3 × 17 + 2 17 + 2 .
Simplify the denominator using the difference of squares: ( 17 − 2 ) ( 17 + 2 ) = 17 − 2 = 15 .
The fraction with a rational denominator is obtained by multiplying by 17 + 2 17 + 2 , so the answer 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 the numerator and denominator by the conjugate of the denominator. The conjugate of 17 − 2 is 17 + 2 .
Multiplying by the Conjugate Therefore, we need to multiply by the fraction 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.
Resulting Fraction So the fraction becomes: 15 3 ( 17 + 2 ) = 5 17 + 2 . The denominator is now rational.
Final Answer Therefore, the correct fraction to multiply by is 17 + 2 17 + 2 .
Examples
Rationalizing the denominator is a technique used in various fields, such as physics and engineering, when dealing with equations involving radicals. For example, when calculating impedance in electrical circuits or when simplifying expressions in quantum mechanics, it is often necessary to eliminate radicals from the denominator to make calculations easier and more accurate. By rationalizing the denominator, we can manipulate equations into a more manageable form, allowing for easier analysis and problem-solving.
To obtain a fraction with a rational denominator from 17 − 2 3 , we multiply by its conjugate 17 + 2 17 + 2 . This operation rationalizes the denominator resulting in a simpler form. Therefore, the answer is option B.
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