Rationalize the denominator and simplify: [tex]$\frac{4-\sqrt{3}}{\sqrt{5}}$[/tex]
Answer (2)
Multiply the numerator and denominator by 5 to rationalize the denominator.
Distribute 5 in the numerator: 4 5 − 3 5 .
Simplify the numerator: 4 5 − 15 .
The final simplified expression is 5 4 5 − 15 .
Explanation
Understanding the Problem We are given the expression 5 4 − 3 . Our goal is to rationalize the denominator, meaning we want to eliminate the square root from the denominator.
Rationalizing the Denominator To rationalize the denominator, we multiply both the numerator and the denominator by 5 . This gives us: 5 4 − 3 × 5 5 = 5 × 5 ( 4 − 3 ) 5 .
Simplifying the Expression Now, we simplify the expression. In the numerator, we distribute 5 : 5 4 5 − 3 5 . Since 3 5 = 3 × 5 = 15 , we have: 5 4 5 − 15 .
Final Answer The simplified expression with a rationalized denominator is 5 4 5 − 15 .
Examples
Rationalizing the denominator is a useful skill in various mathematical contexts. For example, when dealing with impedance in electrical circuits, you might encounter complex numbers with irrational denominators. Rationalizing the denominator helps in simplifying the expression and making it easier to work with. Similarly, in optics, when calculating reflection and transmission coefficients, rationalizing the denominator can simplify the calculations and provide a clearer understanding of the physical quantities involved. This technique is also helpful in simplifying expressions in thermodynamics and fluid mechanics, where irrational numbers often arise in calculations.
To rationalize the denominator of 5 4 − 3 , multiply both the numerator and denominator by 5 to obtain 5 4 5 − 15 . This process removes the square root from the denominator and simplifies the expression. The final answer is 5 4 5 − 15 .
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