Given that [tex]$\frac{\sqrt{3}+\sqrt{5}}{\sqrt{5}}=x+y \sqrt{15}$[/tex], find the value of $(x+y)$.
Answer (2)
By simplifying the expression on the left side and rationalizing the denominator, we determine that x = 1 and y = 5 1 . Therefore, the value of x + y is 5 6 .
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Simplify the left side of the equation: 5 3 + 5 = 5 3 + 1 .
Rationalize the denominator: 5 3 = 5 15 .
Rewrite the equation: 5 15 + 1 = x + y 15 .
Find x and y : x = 1 and y = 5 1 , so x + y = 5 6 .
The final answer is 5 6
Explanation
Problem Analysis We are given the equation 5 3 + 5 = x + y 15 and we need to find the value of x + y .
Simplifying the Left Side First, let's simplify the left side of the equation: 5 3 + 5 = 5 3 + 5 5 = 5 3 + 1
Rationalizing the Denominator Now, we need to rationalize the denominator of the term 5 3 :
5 3 = 5 3 ⋅ 5 5 = 5 3 5 = 5 15
Substituting Back into the Equation Substitute this back into the equation: 5 15 + 1 = x + y 15
Equating Rational and Irrational Parts Now, we can equate the rational and irrational parts of the equation. We have: x = 1 and y = 5 1
Calculating x+y Finally, we can find the value of x + y :
x + y = 1 + 5 1 = 5 5 + 5 1 = 5 6 So, x + y = 5 6 = 1.2
Examples
This type of problem, involving simplification and rationalization of expressions with square roots, is often encountered in fields like physics and engineering when dealing with wave equations or signal processing. For instance, when calculating the impedance of a circuit or analyzing the propagation of electromagnetic waves, you might need to manipulate expressions containing square roots to obtain a simplified form that allows for easier computation and interpretation. Understanding how to rationalize denominators and combine like terms is crucial for accurate calculations and effective problem-solving in these areas.
