$\frac{5}{\sqrt{2}+7}$
Answer (2)
To simplify 2 + 7 5 , we rationalize the denominator by multiplying by its conjugate, resulting in the final expression 47 35 − 5 2 .
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Identify the conjugate of the denominator: The conjugate of 2 + 7 is 2 − 7 .
Multiply the numerator and denominator by the conjugate: 2 + 7 5 × 2 − 7 2 − 7 .
Simplify the expression: 2 − 49 5 ( 2 − 7 ) = − 47 5 2 − 35 .
Multiply by -1 to obtain the final simplified form: 47 35 − 5 2 .
Explanation
Understanding the Problem We are given the expression 2 + 7 5 . Our goal is to simplify this expression by rationalizing the denominator. This means we want to get rid of the square root in the denominator.
Finding the Conjugate To rationalize the denominator, we multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of 2 + 7 is 2 − 7 .
Multiplying the Numerator Multiply the numerator by the conjugate: 5 ( 2 − 7 ) = 5 2 − 35
Multiplying the Denominator Multiply the denominator by the conjugate: ( 2 + 7 ) ( 2 − 7 ) = ( 2 ) 2 − 7 2 = 2 − 49 = − 47
Simplifying the Expression Now, we write the simplified expression: − 47 5 2 − 35 To make the expression look nicer, we can multiply both the numerator and the denominator by -1: − 1 ( − 47 ) − 1 ( 5 2 − 35 ) = 47 35 − 5 2
Final Answer Therefore, the simplified expression is 47 35 − 5 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 calculating forces in mechanics, you might encounter expressions with radicals in the denominator. Rationalizing the denominator helps in standardizing the form of the expression, making it easier to compare and combine with other terms. Consider an electrical circuit where the impedance is given by a complex number with a radical in the denominator; rationalizing simplifies the impedance for further analysis.
