# Multiplication by the conjugate quantity: mult_conjugate

### 5.12.8 Multiplication by the conjugate quantity: mult_conjugate

The mult_conjugate tries to remove square roots from the bottom of an expression.

- mult_conjugate takes one argument:
  expr, an expression. The denominator or numerator is supposed to contain a square root.
- mult_conjugate(expr) returns the following:
  - If expr is a fraction and the denominator contains a square root, then this expression is returned with the numerator and denominator multiplied by the conjugate of the denominator.
  - If expr is a fraction and the numerator, but not the denominator, contains a square root (if expr is not a fraction, it is regarded as a fraction with a denominator of 1), then this expression is returned with the numerator and denominator multiplied by the conjugate of the numerator.

#### Examples.

- Input:
  mult_conjugate((2+sqrt(2))/(2+sqrt(3)))
  Output:

  ⎛
  ⎜
  ⎝

  2+

  √

  2

  ⎞
  ⎟
  ⎠

  ⎛
  ⎜
  ⎝

  2−

  √

  3

  ⎞
  ⎟
  ⎠

  ⎛
  ⎜
  ⎝

  2+

  √

  3

  ⎞
  ⎟
  ⎠

  ⎛
  ⎜
  ⎝

  2−

  √

  3

  ⎞
  ⎟
  ⎠

- Input:
  mult_conjugate((2+sqrt(2))/(sqrt(2)+sqrt(3)))
  Output:

  ⎛
  ⎜
  ⎝

  2+

  √

  2

  ⎞
  ⎟
  ⎠

  ⎛
  ⎜
  ⎝

  −

  √

  2

  +

  √

  3

  ⎞
  ⎟
  ⎠

  ⎛
  ⎜
  ⎝

  √

  2

  +

  √

  3

  ⎞
  ⎟
  ⎠

  ⎛
  ⎜
  ⎝

  −

  √

  2

  +

  √

  3

  ⎞
  ⎟
  ⎠

- Input:
  mult_conjugate((2+sqrt(2))/2)
  Output:

  ⎛
  ⎜
  ⎝

  2+

  √

  2

  ⎞
  ⎟
  ⎠

  ⎛
  ⎜
  ⎝

  2−

  √

  2

  ⎞
  ⎟
  ⎠

  2

  ⎛
  ⎜
  ⎝

  2−

  √

  2

  ⎞
  ⎟
  ⎠