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 17 − 2 3 , we multiply by its conjugate 17 + s q r t 2 17 + s q r t 2 . The correct option that provides an equivalent fraction with a rational denominator is B . This approach simplifies the expression by removing the square roots from the denominator.
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Identify the denominator: 17 − 2 .
Find the conjugate of the denominator: 17 + 2 .
Multiply the given fraction by a fraction consisting of the conjugate divided by itself: 17 + 2 17 + 2 .
The correct fraction 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 should multiply by the fraction 17 + 2 17 + 2 . Let's verify this: 17 − 2 3 × 17 + 2 17 + 2 = ( 17 − 2 ) ( 17 + 2 ) 3 ( 17 + 2 ) .
Simplifying the Denominator The denominator simplifies to ( 17 ) 2 − ( 2 ) 2 = 17 − 2 = 15 , which is a rational number. Thus, the correct fraction is 17 + 2 17 + 2 .
Final Answer The fraction that will produce an equivalent fraction with a rational denominator is 17 + 2 17 + 2 .
Examples
Rationalizing the denominator is a technique used in various fields, such as physics and engineering, to simplify expressions and make calculations easier. For example, when dealing with impedances in electrical circuits or wave functions in quantum mechanics, rationalizing the denominator can help in obtaining more manageable and interpretable results. Imagine you are designing a bridge and need to calculate the stress on a particular support beam. The stress calculation might involve a fraction with a radical in the denominator. By rationalizing the denominator, you can simplify the calculation and obtain a more accurate result, ensuring the bridge's safety and stability.
