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Question 13
Rationalise the denominator in each of the following and hence evaluate by taking 2=1.414,3=1.732 and 5=2.236 upto three places of decimal.
(i)43(ii)66(iii)1052(iv)22+2(v)13+2

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Solution

(i) Let E=43
For rationalising the denominator, multiply numerator and denominator by 3. We get,
E=43×33=433=43×1.732=6.39283=2.309 [Put 3=1.732]

(ii) For rationalising denominator, multiply numerator and denominator by 6. We get,
E=66×66=666=2×3=1.414×1.732=2.449 [Put 2=1.414 and 3=1.732]

(iii) Let E=1052=5252=5(21)2 [10=25]
=2.236(1.4141)2=1.118×0.414=0.462850.463

(iv) Let E=22+2
For rationalizing the denominator, multiply numerator and denominator by 22.
We get,
=22+2×2222=2(22)(2)2(2)2
[Using identity, (ab)(a+b)=a2b2]
=2×2(21)2=2(21)2=21=1.414=0.414 [Put2=1.414]

(v)
Let E=13+2
For rationalising the denominator multiply numerator and denominator by 32.
We get,
13+2×3232=32(3)2(2)2
[Using identity,(ab)(a+b)=a2b2]
=3232=32
=1.7321.414=0.318 [Put 3=1.732 and 2=1.414]

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