Listen
Write a factor that you can use to rationalize the denominator of $\frac{3 m}{\sqrt[3]{4}}$.
A rationalizing factor is $\square$.
Answer (1)
The problem requires finding a factor to rationalize the denominator of 3 4 3 m .
Rewrite the denominator as 3 4 = 2 3 2 .
Determine the rationalizing factor by finding x such that 2 3 2 × 2 x = 2 1 , which gives x = 3 1 .
The rationalizing factor is therefore 3 2 .
3 2
Explanation
Understanding the Problem We are given the fraction 3 4 3 m and we want to find a factor that will rationalize the denominator. This means we want to multiply both the numerator and denominator by a factor that will eliminate the cube root in the denominator.
Analyzing the Denominator The denominator is 3 4 . We can rewrite 4 as 2 2 , so the denominator is 3 2 2 = 2 3 2 . To rationalize the denominator, we need to multiply it by a factor that will result in an integer power of 2.
Finding the Rationalizing Factor We want to find a value x such that 2 3 2 ⋅ 2 x = 2 n , where n is an integer. This means that 3 2 + x = n , so x = n − 3 2 . The smallest positive value for x occurs when n = 1 , so x = 1 − 3 2 = 3 1 . Therefore, the rationalizing factor is 2 3 1 = 3 2 .
Verification To verify, we can multiply the original denominator by our rationalizing factor: 3 4 ⋅ 3 2 = 3 4 ⋅ 2 = 3 8 = 2 , which is a rational number.
Conclusion Thus, the rationalizing factor is 3 2 .
Examples
Rationalizing the denominator is a technique used to simplify expressions and make them easier to work with, especially when performing further calculations or comparisons. For example, in physics, if you are calculating the impedance of a circuit and end up with a complex number in the denominator, you would rationalize it to separate the real and imaginary parts, making it easier to analyze the circuit's behavior. Similarly, in chemistry, when dealing with concentrations or equilibrium constants, rationalizing denominators can help in simplifying calculations and understanding the relationships between different variables.
