Rationalize the denominator and simplify.
$\frac{9}{\sqrt{3}}$
Answer (1)
Multiply the numerator and denominator by 3 to rationalize the denominator.
Simplify the expression: 3 9 3 .
Divide both the numerator and the denominator by 3.
The simplified expression is 3 3 .
Explanation
Understanding the Problem We are given the expression 3 9 . Our goal is to rationalize the denominator, which means we want to eliminate the square root from the denominator.
Rationalizing the Denominator To rationalize the denominator, we multiply both the numerator and the denominator by 3 : 3 9 × 3 3 = 3 × 3 9 3 Remember that 3 × 3 = 3 , so we have: 3 9 3
Simplifying the Expression Now we simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 3: 3 9 3 = 3 9 × 3 = 3 3
Final Answer Therefore, the simplified expression with a rationalized denominator is 3 3 .
Examples
Rationalizing the denominator is a useful technique in various mathematical and scientific contexts. For example, when calculating the impedance of an electrical circuit involving complex numbers, you often need to rationalize the denominator to express the impedance in a standard form. Similarly, in physics, when dealing with wave functions or quantum mechanical calculations, rationalizing the denominator can simplify expressions and make them easier to interpret. This technique ensures that the result is expressed in a clear and conventional manner, facilitating further calculations and analysis.
