The problem is to rationalize the denominator of the fraction 4/(2 - 3√2). The general strategy to rationalize the denominator involves removing any radicals that are fringing the denominator, often by multiplying by something of equal value to not change the overall value but to rid the denominator of the radical.
The first step to rationalize the denominator is to determine the conjugate of the denominator. The conjugate of a binomial (two-termed expression) changes the sign between the terms only. The denominator of our fraction is 2 - 3√2, so its conjugate is 2 + 3√2.
Next, we multiply the numerator and denominator of our fraction by this conjugate (2 + 3√2). Remember, this doesn’t change the value of the fraction because this is the same as multiplying by a form of 1. Algebraically, the structure of this operation looks like this:
(4/(2 - 3√2)) * ((2 + 3√2)/(2 + 3√2))
The numerator multiplication simplifies to: 4 * (2 + 3√2), and the denominator multiplication simplifies to: (2 - 3√2) * (2 + 3√2).
The result of this multiplication simplifies to a new fraction, with values that we need to calculate.
To find out the value of our new numerator, we need to calculate: 4 * (2 + 3√2). This gives 24.97 approximately.
In the same way, calculating the new denominator gives: (2 - 3√2) * (2 + 3√2) which simplifies to -14 approximately.
So, the final fraction with the rationalized denominator is approximately 24.97/-14.