Operation on Rational & Irrational Numbers | Q5

Number System | Extra Questions For Class 9 Maths Chapter 1 | Polynomial expansion

Question 5: Simplify the following:
(a) \({8 + \sqrt{5})}) \({8 - \sqrt{5})})
(b) \({10 + \sqrt{3})}) \({6 + \sqrt{2})})
(c) \{(\sqrt {3} + \sqrt {11})}^2) + \{(\sqrt {3} - \sqrt {11})}^2)


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Explanatory Answer | Chapter Extra Question 5

(a) \({8 + \sqrt{5})}) \({8 - \sqrt{5})})
The expression is of the form (x + y) (x – y) = x2 – y2
So, \({8 + \sqrt{5})}) \({8 - \sqrt{5})} = ({8^2} - \sqrt{5}^2))
= 64 – 5 = 59


(b) \({10 + \sqrt{3})}) \({6 + \sqrt{2})})
Simplifying in such terms is the same as expanding the terms of the expression
= \{(10 \times 6)} + 10 \times \sqrt{2} + \sqrt{3} \times 6 + \sqrt{2} \times \sqrt{3})
= \60 + 10\sqrt{2} + 6\sqrt{3} + \sqrt{6})


(c) \{(\sqrt {3} + \sqrt {11})}^2 + {(\sqrt {3} - \sqrt {11})}^2)

\{(\sqrt {3} + \sqrt {11})}^2) = \{(\sqrt {3})}^2) + \{(\sqrt {11})}^2 + 2 \times \sqrt{3} \times \sqrt{11})
\= 3 + 11 + 2\sqrt{33} = 14 + 2\sqrt{33})

\{(\sqrt {3} - \sqrt {11})}^2) = \{(\sqrt {3})}^2) + \{(\sqrt {11})}^2 - 2 \times \sqrt{3} \times \sqrt{11})
\= 3 + 11 - 2\sqrt{33} = 14 - 2\sqrt{33})

Thus, \{(\sqrt {3} + \sqrt {11})}^2) + \{(\sqrt {3} - \sqrt {11})}^2 = 14 + 2\sqrt{33} + 14 - 2\sqrt{33})
= 28

 


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