Rationalize the denominator and simplify: [tex]$\sqrt{\frac{5}{2}}$[/tex]
Answer (1)
Rewrite the expression as a fraction of square roots: 2 5 = 2 5 .
Multiply the numerator and denominator by 2 to rationalize the denominator: 2 5 ⋅ 2 2 = 2 × 2 5 × 2 .
Simplify the expression: 2 × 2 5 × 2 = 2 10 .
The rationalized and simplified form of the expression is: 2 10 .
Explanation
Understanding the Problem We are asked to rationalize the denominator and simplify the expression 2 5 . Rationalizing the denominator means eliminating the square root from the denominator. Simplifying means writing the expression in its simplest form.
Rewriting the Expression We can rewrite the expression as a fraction of square roots: 2 5 = 2 5
Rationalizing the Denominator To rationalize the denominator, we multiply both the numerator and the denominator by 2 : 2 5 ⋅ 2 2 = 2 × 2 5 × 2
Simplifying the Expression Now, we simplify the expression. Recall that a × b = a × b and a × a = a . Therefore, 2 × 2 5 × 2 = 2 5 × 2 = 2 10
Final Answer The expression is now simplified and the denominator is rationalized. The final simplified expression is 2 10 .
Examples
Imagine you're a chef and a recipe calls for a specific amount of an ingredient, but the measuring tool you have expresses it in a form that's hard to work with directly, like 2 5 cups. Rationalizing the denominator, as we did here, transforms the measurement into a more practical form, 2 10 cups, making it easier to measure accurately. This technique is useful not only in cooking but also in various fields where precise measurements and simplified expressions are essential for calculations and practical applications.
