The easiest way to solve calculations involving roots, such as the expression $$\frac{\frac{1}{2}}{1-\frac{\sqrt{2}}{2}}$$, is to first find a common denominator and rationalize the denominator. By multiplying the fraction by $$1=\frac{2}{2}$$, the expression simplifies to $$\frac{1}{2-\sqrt{2}}$$. Further rationalization using the difference of squares formula leads to the final result of $$\frac{2+\sqrt{2}}{2}$$ or $$1+\frac{\sqrt{2}}{2}$. This method is effective for similar root calculations.
PREREQUISITESStudents, educators, and anyone interested in mastering algebraic techniques for solving expressions involving roots and fractions.
wishmaster said:Im just wondering what is the easiest way to deal with calculations where roots are involved?
For example how do you solve this one?
$$\frac{\frac{1}{2}}{1-\frac{\sqrt{2}}{2}}$$Thank you for replies!
Prove It said:A rule of thumb for fractions is to get a common denominator and to always attempt to rationalise denominators.
What have you tried so far?
wishmaster said:To multiply the fraction with 2.
MarkFL said:I assume you mean to multiply by:
$$1=\frac{2}{2}$$
This is a good first step. :D What did you get in doing so?
wishmaster said:Yes,Mark,i did it so as you said. Actualy i understand this,but i am wondering if there some other way to deal with this...
And i got:
$$\frac{1}{2-\sqrt{2}}$$
MarkFL said:Okay, now you want to rationalize the denominator. Think of the difference of squares formula...
wishmaster said:I multiply fraction with $$(2+\sqrt{2})$$
So i got:
$$\frac{2+\sqrt{2}}{2}$$
thank you!MarkFL said:Good! You could choose to leave it like that, or express it as:
$$1+\frac{\sqrt{2}}{2}$$