Transformation of Irrational Expressions

For transformation of irrational expressions we use the properties of radicals and the properties of powers with rational exponents.

Example. Simlify the following expression:`f(x)=((root(4)(x^3)-root(4)(x))/(1-sqrt(x))+(1+sqrt(x))/root(4)(x))^2*(X^0+2/sqrt(x)+x^(-1))^(-1/2)`.

  1. `(root(4)(x^3)-root(4)(x))/(1-sqrt(x))=(root(4)(x)(root(4)(x^2)-1))/(1-sqrt(x))=(root(4)(x)(sqrt(x)-1))/(1-sqrt(x))=-root(4)(x)`;
  2. `-root(4)(x)+(1+sqrt(x))/root(4)(x)=(-root(4)(x)*root(4)(x)+1+sqrt(x))/root(4)(x)=(-sqrt(x)+1+sqrt(x))/root(4)(x)=1/root(4)(x)`;
  3. `(1/root(4)(x))^2=1/(root(4)(x))^2=1/sqrt(x)`;
  4. `x^0+2/sqrt(x)+x^(-1)=1+2/sqrt(x)+1/x=(x+2sqrt(x)+1)/x=(sqrt(x)+1)^2/x`;
  5. `(((sqrt(x)+1)^2)/x)^(-1/2)=(x/(sqrt(x)+1)^2)^(1/2)=sqrt(x/(sqrt(x)+1)^2)=sqrt(x)/(sqrt(x)+1)`;
  6. `1/sqrt(x)*sqrt(x)/(sqrt(x)+1)=1/(sqrt(x)+1)`.

We should write the answer in such way, that there is no irrationality in the denominator. To get rid of irrationality in denominator of fraction `1/(sqrt(x)+1)` we multiply the numerator and denominator by `sqrt(x)-1` - this expression is called conjugate for expression `sqrt(x)+1`. Then we will obtain `1/(sqrt(x)+1)=(sqrt(x)-1)/((sqrt(x)+1)(sqrt(x)-1))=(sqrt(x)-1)/((sqrt(x))^2-1^2)=(sqrt(x)-1)/(x-1)`.