Rationalize the denominator and simplify.
[tex]$\sqrt{\frac{5}{10}}$[/tex]
Answer (1)
Simplify the fraction inside the square root: 10 5 = 2 1 , so the expression becomes 2 1 .
Rewrite the expression as a fraction of square roots: 2 1 = 2 1 = 2 1 .
Rationalize the denominator by multiplying the numerator and denominator by 2 : 2 1 ⋅ 2 2 = 2 2 .
The rationalized and simplified form is: 2 2 .
Explanation
Understanding the Problem We are given the expression 10 5 . Our goal is to rationalize the denominator and simplify the expression.
Simplifying the Fraction First, we simplify the fraction inside the square root: 10 5 = 2 1 So the expression becomes 2 1 .
Separating the Square Root Next, we rewrite the square root of a fraction as a fraction of square roots: 2 1 = 2 1 = 2 1
Rationalizing the Denominator To rationalize the denominator, we multiply both the numerator and the denominator by 2 : 2 1 ⋅ 2 2 = 2 2 Thus, the simplified expression is 2 2 .
Final Answer Therefore, the rationalized and simplified form of 10 5 is 2 2 .
Examples
Rationalizing the denominator is a useful skill in various fields, such as physics and engineering, where simplified expressions are preferred for calculations. For example, when dealing with impedance in electrical circuits or calculating forces in mechanics, rationalizing denominators can make the calculations easier and more accurate. Consider a scenario where you need to calculate the current in a circuit involving a complex impedance. If the impedance involves a square root in the denominator, rationalizing it simplifies the expression and allows for easier computation of the current using Ohm's law.
