[Clean the question above and then proceed with classification]
$\frac{1}{1-\sqrt{7}}$
$1+\sqrt{7}$
Answer (2)
Rationalize the denominator of 1 − 7 1 by multiplying the numerator and denominator by 1 + 7 .
Simplify the denominator using the difference of squares: ( 1 − 7 ) ( 1 + 7 ) = 1 − 7 = − 6 .
The expression becomes − 6 1 + 7 .
The simplified expression is − 6 1 + 7 .
Explanation
Problem Analysis We are given two expressions: 1 − 7 1 and 1 + 7 . Our goal is to rationalize the denominator of the first expression and see if it is related to the second expression.
Rationalizing the Denominator To rationalize the denominator of 1 − 7 1 , we multiply both the numerator and the denominator by the conjugate of the denominator, which is 1 + 7 . This gives us: 1 − 7 1 × 1 + 7 1 + 7 = ( 1 − 7 ) ( 1 + 7 ) 1 + 7 .
Simplifying the Expression Now, we simplify the denominator: ( 1 − 7 ) ( 1 + 7 ) = 1 2 − ( 7 ) 2 = 1 − 7 = − 6. So, the expression becomes: − 6 1 + 7 = − 6 1 + 7 .
Comparing the Expressions Comparing the simplified expression − 6 1 + 7 with the second given expression 1 + 7 , we see that the first expression is equal to the second expression multiplied by − 6 1 .
Final Answer Therefore, the expression 1 − 7 1 simplifies to − 6 1 + 7 .
Examples
Rationalizing the denominator is a useful technique in various mathematical and scientific contexts. For example, when dealing with electrical circuits, you might encounter complex impedances involving square roots in the denominator. Rationalizing the denominator helps simplify calculations and allows for easier comparison and analysis of circuit parameters. Similarly, in optics, when calculating refractive indices or dealing with wave interference, rationalizing denominators can lead to more manageable expressions.
To simplify 1 − 7 1 , multiply by the conjugate 1 + 7 to get − 6 1 + 7 . This rationalizes the denominator and shows the connection to the expression 1 + 7 .
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