1.15 Rationalizing Binomial Denominators

• The binomial numbers $\sqrt{A}+\sqrt{B}$ and $\sqrt{A}-\sqrt{B}$ are called conjugates of each other.

  For example, $\sqrt{2}+\sqrt{5}$ and $\sqrt{2}-\sqrt{5},$ or $\sqrt{3x+5}+\sqrt{2x+7}$ and $\sqrt{3x+5}-\sqrt{2x+7}$ are conjugates of each other.

• Because conjugates are the sum and difference of the same two terms, their product is the difference of the squares of these terms (see  Special Product Formulas); that is, $$(\sqrt{A}-\sqrt{B})(\sqrt{A}+\sqrt{B})=(\sqrt{A}{)}^2-(\sqrt{B}{)}^2=A-B.$$

• Remark that $C\sqrt{A}-D\sqrt{B}$ and $C\sqrt{A}+D\sqrt{B}$ are conjugates of each other.

• If the denominator of a fraction is of the form $\sqrt{A}-\sqrt{B}$ (or $C\sqrt{A}-D\sqrt{B}$), we can rationalize the denominator by multiplying the numerator and denominator of the fraction by the conjugate $\sqrt{A}+\sqrt{B}$ (or $C\sqrt{A}+D\sqrt{B}$).

  For example,
  $$\begin{array}{rl}\frac{13}{\sqrt{5}+3\sqrt{2}}& =\frac{13}{\sqrt{5}+3\sqrt{2}}\cdot \frac{\sqrt{5}-3\sqrt{2}}{\sqrt{5}-3\sqrt{2}}\\ & =\frac{13(\sqrt{5}-3\sqrt{2})}{5-3^2\times 2}\\ & =-(\sqrt{5}-3\sqrt{2})\\ & =3\sqrt{2}-\sqrt{5}\end{array}$$

Similarly

• If the denominator of a fraction is $\sqrt[3]{A}+\sqrt[3]{B}$, we multiply the numerator and denominator of the fraction by $\sqrt[3]{A^2}-\sqrt[3]{AB}+\sqrt[3]{B^2}$ and use the Sum of Cubes formula to get a denominator of $A+B$ (see Special Product Formulas)

• If the denominator of a fraction is $\sqrt[3]{A}-\sqrt[3]{B}$, we multiply the numerator and denominator of the fraction by $\sqrt[3]{A^2}+\sqrt[3]{AB}+\sqrt[3]{B^2}$ and use the Difference of Cubes formula to get a denominator of $A-B$ (see Special Product Formulas).

  For example:
  $$\begin{array}{rl}\frac{5}{\sqrt[3]{3}-2}& =\frac{5}{\sqrt[3]{3}-\sqrt[3]{2^3}}\cdot \frac{\sqrt[3]{9}+\sqrt[3]{3\times 2^3}+\sqrt[3]{2^6}}{\sqrt[3]{9}+\sqrt[3]{3\times 2^3}+\sqrt[3]{2^6}}\\ & =\frac{5}{3-2^3}\cdot (\sqrt[3]{9}+\sqrt[3]{3\times 2^3}+\sqrt[3]{2^6})\\ & =-\sqrt[3]{9}-\sqrt[3]{2^3\times 3}-2^{6/3}\\ & =-\sqrt[3]{9}-2\sqrt[3]{3}-4.\end{array}$$