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

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 = $√^4{5}$

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: (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: (c)

$(2.89)^{0.5}, 2 - (0.5)^2, √3$ and $√^3{0.008}$

$(2.89)^{0.5} = (2.89)^{1/2}$ =1.7,

$2 - (0.5)^2$= 2 - 0.25 = 1.75,

$√3$= 1.732 and

$√^3{0.008}= √^3{0.2 ×0.2 ×0.2}$=0.2

Obviously,

0.2 < 1.7 < 1.732 < 1.75

$√^3{0.008}<(2.89)^{0.5}<√3<2 - (0.5)^2$

Answer: (c)

$2^60, 3^48, 4^36$ and $5^24$

$2^60 = (2^5)^12 =(32)^12$

$5^24 = (5^2)^12 =(25)^12$

$2^60 >5^24$

$3^48 =(3^4)^12 =(81)^12$

$3^48 >2^60$

$4^36 =(4^3)^12 = (64)^12$

$3^48$ is the largest number

Answer: (d)

$√{0.09}, √^3{0.064},$ 0.5 and $3/5$

$√{0.09}=0.3$

$√^3{0.064}=0.4$; 0.5;

$3/5$= 0.6

Clearly, $√{0.09}<√^3{0.064}<0.5<3/5$