model 1 find largest and smallest value Section-Wise Topic Notes With Detailed Explanation And Example Questions

Answer: (d)

$(√{19} - √{17}),(√{13} -√{11}),(√7 -√5)$ and $(√5-√3)$

$√{19} - √{17}$

=${(√19-√17)×(√19+√17)}/{√19+√17}$

${19-17}/{√19+√17}=2/{√19+√17}$

Similarly,$√{13} -√{11}=2/{√{13} +√{11}}$

$√7 -√5=2/{√7 +√5}$

$√5-√3=2/{√5+√3}$

Clearly, $√5-√3$ is the greatest.

(Smaller the denominator, greater the no.)

Answer: (c)

$√2, √^3{9}, √^4{16}, √^5{32}$

$(16)^{1/4}=(2^4)^{1/4}=2$

$√^5{32}=(32)^{1/5}=(2^5)^{1/5}=2$

$√^3{9}>2,√2<2$

Answer: (a)

$√5 , 3√7 , 4√13$

$√5$

$3√7 =√{9×7}=√63$

$√^4{13}=√{4×4×13}=√{208}$

Clearly,$√5<3√7<4√13$

Answer: (c)

$√2, √^3{3}, √^6{6}, √^5{5}$

LCM of 2, 3, 6 & 5 = 30

$2^{1/2} = 2^{15/30} = √^30{2^15}=32768$

$3^{1/3} = 3^{10/30} = √^30{3^10}= 59049$

$6^{1/6} = 6^{5/30} = √^30{6^5}=7776$

$5^{1/5} = 5^{6/30} = √^30{5^5}$= 15625

Therefore, $√^3{3}$ is the greatest.

Answer: (b)

$√^3{4}, √2, √^6{3}, √^4{5}$

L.C.M. of 3, 2, 6, 4, = 12

$√^3{4}=(4)^{1/3}=(4)^{4/12}$

=$(4^4)^{1/12}=(256)^{1/12}$

$√2 =(2)^{1/2}=(2)^{6/12}$

=$(2^6)^{1/12}=(64)^{1/12}$

$√^6{3}=(3)^{1/6}=(3)^{2/12}=(3^2)^{1/12}=(9)^{1/12}$

$√^4{5}=(5)^{1/4}=(5)^{3/12}=(5^3)^{1/12}=(125)^{1/12}$

$(256)^{1/12}>(125)^{1/12}>(64)^{1/12}>(9)^{1/12}$

or, $√^3{4}>√^4{5}>√2>√^6{3}$

Answer: (d)

$√^6{12}, √^3{4}, √^4{5}, √3$

LCM of indices of surds

= LCM of 6, 3, 4 and 2 = 12

$√^6{12} =√^12{2^2}=√^12{144}$

$√^3{4} =√^12{4^4}=√^12{256}$

$√^4{5} =√^12{5^3}=√^12{125}$

$√3 =√^12{3^6}=√^12{729}$

The smallest surd = $√^4{5}$