Rationalize the denominator and simplify. [tex]$\sqrt{\frac{3}{5}}$[/tex]
Answer (1)
Rewrite the square root of the fraction as a fraction of square roots: 5 3 = 5 3 .
Multiply the numerator and denominator by 5 to rationalize the denominator: 5 3 ⋅ 5 5 .
Simplify the expression: 5 ⋅ 5 3 ⋅ 5 = 5 15 .
The final simplified expression with a rationalized denominator is: 5 15 .
Explanation
Understanding the problem We are asked to rationalize the denominator and simplify the expression 5 3 . This means we want to rewrite the expression so that there are no square roots in the denominator.
Separating the square root First, we can rewrite the square root of a fraction as a fraction of square roots: 5 3 = 5 3 .
Rationalizing the denominator To rationalize the denominator, we need to get rid of the 5 in the denominator. We can do this by multiplying both the numerator and the denominator by 5 : 5 3 ⋅ 5 5 = 5 ⋅ 5 3 ⋅ 5 .
Simplifying the expression Now, we simplify the expression. In the numerator, we have 3 ⋅ 5 = 3 × 5 = 15 . In the denominator, we have 5 ⋅ 5 = 5 . So the expression becomes: 5 15 .
Final Answer The expression is now simplified, and the denominator is rationalized. Therefore, the final answer is 5 15 .
Examples
Rationalizing the denominator is a useful skill in various areas of mathematics, such as trigonometry and calculus. For example, when dealing with trigonometric functions, you might encounter expressions like 2 1 , which is often rewritten as 2 2 for easier manipulation. In calculus, rationalizing the denominator can help simplify expressions before integration or differentiation, making the problem easier to solve. This technique is also useful in real-world applications, such as physics and engineering, where simplifying expressions can lead to more accurate calculations and better understanding of the problem.
