Rationalize the denominator and simplify: [tex]$\frac{4-\sqrt{3}}{\sqrt{5}}$[/tex]
Answer (1)
Multiply the numerator and denominator by 5 .
Distribute 5 in the numerator: 4 5 − 15 .
Simplify the denominator: 5 × 5 = 5 .
The rationalized and 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 : 5 4 − 3 × 5 5 .
Performing the Multiplication Multiplying the numerator, we get ( 4 − 3 ) 5 = 4 5 − 3 5 = 4 5 − 15 . The denominator becomes 5 × 5 = 5 .
Final Result Therefore, the simplified expression is 5 4 5 − 15 .
Examples
Rationalizing the denominator is a technique used in various fields, such as physics and engineering, to simplify calculations and make expressions easier to work with. For example, when calculating impedance in electrical circuits or dealing with wave functions in quantum mechanics, you might encounter expressions with radicals in the denominator. Rationalizing the denominator makes it easier to compare and combine these expressions, leading to more accurate and efficient problem-solving.
