Rationalising the denominator

We rationalise the denominator to ensure that it becomes easier to perform any calculation on the rational number. When we rationalise the denominator in a fraction, then we are eliminating any radical expressions such as square roots and cube roots from the denominator.

Definition

Rationalising is the process of multiplying a surd with another similar surd, to get a rational number. The surd that is used to multiply is called the rationalising factor.

• To rationalise ${\sqrt {x}}$ we need another ${\sqrt {x}}$ , ${\sqrt {x}}\times {\sqrt {x}}=x$
• To rationalise $a+{\sqrt {b}}$ we need a rationalising factor $a-{\sqrt {b}}$ , $(a+{\sqrt {b}})\times (a-{\sqrt {b}})=a^{2}-({\sqrt {b}})^{2}=a^{2}-b$
• To rationalise he rationalising factor of 2√3 is √3: 2√3 × √3 = 2 × 3 = 6.

Rationalise the Denominator Meaning

Rationalising the denominator means the process of moving a root, for instance, a cube root or a square root from the bottom of a fraction (denominator) to the top of the fraction (numerator). By this we bring the fraction to its simplest form thereby, the denominator becomes rational.

Examples

1. Rationalise the denominator ${\frac {1}{7+3{\sqrt {2}}}}$

${\frac {1}{7+3{\sqrt {2}}}}={\frac {1}{7+3{\sqrt {2}}}}\times {\frac {(7-3{\sqrt {2}})}{(7-3{\sqrt {2}})}}={\frac {(7-3{\sqrt {2}})}{7^{2}-(3{\sqrt {2}})^{2}}}={\frac {(7-3{\sqrt {2}})}{49-18}}={\frac {(7-3{\sqrt {2}})}{31}}$

2. Rationalise the denominator ${\frac {5}{{\sqrt {3}}-{\sqrt {5}}}}$

${\frac {5}{{\sqrt {3}}-{\sqrt {5}}}}={\frac {5}{{\sqrt {3}}-{\sqrt {5}}}}\times {\frac {({\sqrt {3}}+{\sqrt {5}})}{({\sqrt {3}}+{\sqrt {5}})}}={\frac {5({\sqrt {3}}+{\sqrt {5}})}{({\sqrt {3}})^{2}-({\sqrt {5}})^{2}}}={\frac {5({\sqrt {3}}+{\sqrt {5}})}{3-5}}=-{\frac {5}{2}}({\sqrt {3}}+{\sqrt {5}})$