Rationalize the denominator and simplify.
$\frac{\sqrt{7}}{\sqrt{5}}$
Answer (1)
Multiply the numerator and denominator by 5 to get 5 × 5 7 × 5 .
Simplify the numerator: 7 × 5 = 35 .
Simplify the denominator: 5 × 5 = 5 .
The rationalized and simplified expression is 5 35 .
Explanation
Understanding the Problem We are given the expression 5 7 . Our goal is to rationalize the denominator, which means we want to eliminate the square root from the denominator.
Multiplying by 5 5 To rationalize the denominator, we multiply both the numerator and the denominator by 5 . This gives us: 5 7 × 5 5 = 5 × 5 7 × 5
Simplifying the Expression Now, we simplify the expression. Recall that a × b = a × b and a × a = a . Therefore, we have: 5 × 5 7 × 5 = 5 7 × 5 = 5 35
Final Answer The simplified expression with a rationalized denominator is 5 35 .
Examples
Rationalizing the denominator is a technique used in various mathematical contexts, such as simplifying expressions in algebra, calculus, and trigonometry. For example, when dealing with impedance in electrical engineering, you might encounter complex numbers with square roots in the denominator. Rationalizing the denominator helps in simplifying these expressions, making them easier to work with and interpret. Similarly, in physics, when calculating energy levels or wave functions, rationalizing denominators can lead to more manageable and understandable results.
