Extra Questions For CBSE Class 9 Maths Chapter 1 | Q6

Number Systems | Operation on Real numbers - Rationalize the denominators

Question 6: 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}})


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Explanatory Answer | Number Systems Extra Question 6

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

Multiply and divide by the conjugate of the denominator
The conjugate of (√3 – 1) is (√3 + 1)
\\frac{2}{\sqrt{3} - 1}) × \\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 (√12 – √5) is (√12 + √5)
\\frac{7}{\sqrt{12} - \sqrt{5}}) × \\frac{\sqrt{12} + \sqrt{5}}{{\sqrt12} + {\sqrt5}}) = \\frac{7\sqrt{12} + \sqrt{5}}{\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√5) is (8 - 3√5)
=\\frac{1}{8 + 3\sqrt{5}})× \\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}})

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 – √2 + √5)
\\frac{1}{4 + \sqrt{2} + \sqrt{5}}) × \\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√10)
\\frac{4 - \sqrt{2} - \sqrt{5}}{9 - 2\sqrt{10}}) × \\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})

 


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