Problems Based on Rationalization or Double Rationalization

Rationalization is a technique used in mathematics to eliminate irrational numbers from the denominator of a fraction. This process often involves multiplying the numerator and the denominator by a suitable expression that will make the denominator rational. Double rationalization is a more complex form of rationalization that may be required when dealing with nested radicals or more complicated irrational expressions.

Rationalization

When we have a single square root in the denominator, we can rationalize it by multiplying the numerator and the denominator by the conjugate of the denominator. The conjugate of a binomial (a + b) is (a - b), and vice versa.

Single Rationalization Formula

For a fraction of the form (\frac{a}{\sqrt{b}}), we can rationalize the denominator by multiplying both the numerator and the denominator by (\sqrt{b}):

[ \frac{a}{\sqrt{b}} \times \frac{\sqrt{b}}{\sqrt{b}} = \frac{a\sqrt{b}}{b} ]

Example of Single Rationalization

Rationalize the denominator of (\frac{3}{\sqrt{5}}):

[ \frac{3}{\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}} = \frac{3\sqrt{5}}{5} ]

Double Rationalization

Double rationalization is used when we have a more complex expression, such as a binomial in the denominator that includes a square root or a nested radical (a radical within another radical).

Double Rationalization Formula

For a fraction of the form (\frac{a}{\sqrt{b} + \sqrt{c}}), we can rationalize the denominator by multiplying both the numerator and the denominator by the conjugate (\sqrt{b} - \sqrt{c}):

[ \frac{a}{\sqrt{b} + \sqrt{c}} \times \frac{\sqrt{b} - \sqrt{c}}{\sqrt{b} - \sqrt{c}} = \frac{a(\sqrt{b} - \sqrt{c})}{b - c} ]

Example of Double Rationalization

Rationalize the denominator of (\frac{4}{\sqrt{3} + 1}):

[ \frac{4}{\sqrt{3} + 1} \times \frac{\sqrt{3} - 1}{\sqrt{3} - 1} = \frac{4(\sqrt{3} - 1)}{3 - 1} = \frac{4\sqrt{3} - 4}{2} = 2\sqrt{3} - 2 ]

Table of Differences and Important Points

Aspect	Single Rationalization	Double Rationalization
Purpose	To eliminate a single square root from the denominator	To eliminate a binomial involving square roots from the denominator
Conjugate	Not applicable (only one term in the denominator)	The conjugate of the denominator is used
Complexity	Relatively simple	More complex due to the binomial nature
Formula	(\frac{a}{\sqrt{b}} \times \frac{\sqrt{b}}{\sqrt{b}})	(\frac{a}{\sqrt{b} + \sqrt{c}} \times \frac{\sqrt{b} - \sqrt{c}}{\sqrt{b} - \sqrt{c}})
Result	The denominator becomes a rational number	The denominator becomes a rational number without radicals

Additional Examples

Example 1: Single Rationalization

Rationalize the denominator of (\frac{2}{\sqrt{7}}):

[ \frac{2}{\sqrt{7}} \times \frac{\sqrt{7}}{\sqrt{7}} = \frac{2\sqrt{7}}{7} ]

Example 2: Double Rationalization

Rationalize the denominator of (\frac{5}{\sqrt{2} + \sqrt{3}}):

[ \frac{5}{\sqrt{2} + \sqrt{3}} \times \frac{\sqrt{2} - \sqrt{3}}{\sqrt{2} - \sqrt{3}} = \frac{5(\sqrt{2} - \sqrt{3})}{2 - 3} = -5(\sqrt{2} - \sqrt{3}) ]

Conclusion

Rationalization and double rationalization are important techniques in algebra, especially when simplifying expressions and solving limits in calculus. Understanding how to apply these methods allows for the simplification of irrational numbers, making it easier to perform further mathematical operations and analyses.