Rationalize the denominator and simplify.
$\sqrt{\frac{3}{2}}$
Answer (1)
Rewrite the expression as a fraction of square roots: 2 3 = 2 3 .
Multiply the numerator and denominator by 2 to rationalize the denominator: 2 3 ⋅ 2 2 .
Simplify the numerator: 3 ⋅ 2 = 6 .
Simplify the denominator: 2 ⋅ 2 = 2 . The final answer is 2 6 .
Explanation
Understanding the Problem We are given the expression 2 3 . Our goal is to rationalize the denominator, which means we want to eliminate any square roots from the denominator.
Rewriting the Expression First, we can rewrite the expression as a quotient of square roots: 2 3 = 2 3 .
Rationalizing the Denominator To rationalize the denominator, we multiply both the numerator and the denominator by 2 : 2 3 ⋅ 2 2 = 2 ⋅ 2 3 ⋅ 2 .
Simplifying the Expression Now, we simplify the expression. In the numerator, we have 3 ⋅ 2 = 3 ⋅ 2 = 6 . In the denominator, we have 2 ⋅ 2 = 2 . Therefore, the simplified expression is: 2 6 .
Final Answer Thus, the rationalized and simplified form of 2 3 is 2 6 .
Examples
Rationalizing the denominator is a useful skill in various mathematical contexts. For example, when dealing with impedance in electrical circuits, you often encounter expressions with square roots in the denominator. Rationalizing the denominator helps in simplifying calculations and making the expression easier to work with. Suppose you have an impedance Z = 2 10 ohms. To make it easier to compare with other impedances, you rationalize the denominator: Z = 2 10 × 2 2 = 2 10 2 = 5 2 ohms.
