Final answer:
To rationalize 5/(√7 + 1), multiply both numerator and denominator by the conjugate (√7 - 1), simplify the denominator using the difference of squares, distribute the numerator, and check the result for correctness.
Step-by-step explanation:
To rationalize the denominator of the fraction 5/(√7 + 1), we need to eliminate the square root from the denominator. We can do this by multiplying the numerator and denominator by the conjugate of the denominator. The conjugate of √7 + 1 is √7 - 1 because it is the same two terms but with the opposite sign between them.
Here are the steps to perform this operation:
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- Multiply the numerator and denominator by the conjugate of the denominator, which is √7 - 1.
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- Apply the difference of squares formula to the denominator, which simplifies to 7 - 1 because (√7)^2 = 7 and 1^2 = 1.
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- The product in the denominator is now 6, which is the rational number we were looking for.
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- Finally, distribute the 5 in the numerator to complete the rationalization.
The final rationalized form of the fraction is (5√7 - 5) / 6.
It is important to check the answer to see if it is reasonable and that the original value of the expression has not been changed.