3.4.3 Conjugate

Conjugate multiplies the subject by a unit value consisting of the conjugate of the subject divided by itself. The transformation applies to subjects consisting of addition, subtraction, exponentiation and division. The conjugate is obtained by negating the right operand of the subject, or for division, the right operand with its own right operand negated. The transformations are summarized in the Figure 3.4.

Conjugations are useful when attempting to invert an expression. For example, (x+3)^-n can be turned into 1÷(x+3)^n by transforming the former into (x+3)^-n⋅((x+3)^n÷(x+3)^n) using Conjugate , selecting the denominator and moving it right (x+3)^-n⋅((x+3)^n⋅(1÷(x+3)^n)). This expression must be rectified so the first multiply operator contains operands whose exponents have opposite signs. The rectified expression can then be simplified to (x+3)^(-n+n)÷(x+3)^n and finally to 1÷(x+3)^n.

Subject	Transformation
a+b	(a+b)⋅((a-b)÷(a-b))
a-b	(a-b)⋅((a+b)÷(a+b))
a^b	a^b⋅(a^-b÷a^-b)
a÷(b+c)	a÷(b+c)⋅((b-c)÷(b-c))
a÷(b-c)	a÷(b-c)⋅((b+c)÷(b+c))
a÷(b+c)^n	a÷(b+c)^n⋅((b+c)^-n÷(b+c)^-n)