Number system practice

Operation on real numbers - Rationalisation

Question: Rationalize the denominator:

(a) \\frac{2}{\sqrt{3} - 1})

(b) \\frac{7}{\sqrt{12} - \sqrt{5}})

(c) \\frac{1}{8 + 3\sqrt{5}})

(d) \\frac{1}{4 + \sqrt{2} + \sqrt{5}})

Video Explanation for (a), (b) and (c)

Explanatory Answer

(a) \\frac{2}{\sqrt{3} - 1})

Multiply and divide by the conjugate of the denominator

The conjugate of \\sqrt{3} – 1) is \\sqrt{3} + 1)

\\frac{2}{\sqrt{3} - 1}) \\times) \\frac{{\sqrt{3}} + 1}{{\sqrt{3}} + 1}) = \\frac{{2(\sqrt{3} + 1)}}{{\sqrt{3}^2} - 1^2}) = \\frac{{2(\sqrt{3} + 1)}}{3 - 1})

= \\frac{{2(\sqrt{3} + 1)}}{2}) = \\sqrt{3}) + 1

(b) \\frac{7}{\sqrt{12} - \sqrt{5}})

Multiply and divide by the conjugate of the denominator

The conjugate of \\sqrt{12}) – \\sqrt{5}) is \\sqrt{12}) + \\sqrt{5})

\\frac{7}{\sqrt{12} - \sqrt{5}}) \\times) \\frac{{\sqrt12} + {\sqrt5}}{{\sqrt12} + {\sqrt5}}) = \\frac{7(\sqrt12 + \sqrt5)}{{(\sqrt{12})^2} - {(\sqrt{5})^2}})

= \\frac{7(\sqrt{12} + \sqrt{5})}{12 - 5})

= \\frac{7(\sqrt{12} + \sqrt{5})}{7}) = \\sqrt{12} + \sqrt{5})

(c) \\frac{1}{8 + 3\sqrt{5}})

Multiply and divide the fraction by the conjugate of the denominator

The conjugate of \8 + 3\sqrt{5}) is \8 - 3\sqrt{5})

=\\frac{1}{8 + 3\sqrt{5}}) \\times) \\frac{8 - 3\sqrt{5}}{8 - 3\sqrt{5}}) = \\frac{8 - 3\sqrt{5}}{({8})^2 - ({3\sqrt{5}})^2})

= \\frac{8 - 3\sqrt{5}}{64 - 45}) = \\frac{8 - 3\sqrt{5}}{19})

(d) \\frac{1}{4 + \sqrt{2} + \sqrt{5}})

Video Explanation

This one is more difficult than the previous 3 questions.
Let us do it in 2 steps.

Step 1: Multiply and divide the expression by \4 – {(\sqrt{2} + \sqrt{5})})

\\frac{1}{4 + \sqrt{2} + \sqrt{5}}) \\times) \\frac{4 - (\sqrt{2} + \sqrt{5})}{4 - (\sqrt{2} + \sqrt{5})}) = \\frac{4 - (\sqrt{2} + \sqrt{5})}{(4)^2 - (\sqrt{2} + \sqrt{5}))^2})

= \\frac{4 - (\sqrt{2} + \sqrt{5})}{16 - (2 + 5 + 2\sqrt{10})})

= \\frac{4 - (\sqrt{2} + \sqrt{5})}{9 - 2\sqrt{10}})

Step 2: Multiply and divide the expression by the conjugate of the denominator. i.e., by \9 + 2\sqrt{10})

\\frac{4 - \sqrt{2} + \sqrt{5}}{9 - 2\sqrt{10}} \times \frac{9 + 2\sqrt{10}}{9 + 2\sqrt{10}})

= \\frac{36 + 8\sqrt{10} - 9\sqrt{2} - 2\sqrt{20} - 9\sqrt{5} - 2\sqrt{50}}{81 - 40})

= \\frac{36 + 8\sqrt{10} - 9\sqrt{2} - 4\sqrt{5} - 9\sqrt{5} - 10\sqrt{2}}{41})

= \\frac{36 + 8\sqrt{10} - 19\sqrt{2} - 13\sqrt{5}}{41})