model 1 find largest and smallest value Practice Questions Answers Test with Solutions & More Shortcuts

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: (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: (d)

Making each surd of the same order :

LCM of 3, 4 and 6 = 12

$√^3{9}=(9)^{1/3}=(9)^{4/12}=(9^4)^{1/12}$

$=√^12{9^4}=√^12{6561}$

$√^4{20}=√^12{20^3}=√^12{8000}$

$√^6{25}=√^12{25^2}=√^12{625}$

$√^12{625}<√^12{6561}<√^12{8000}$

$√^6{25}<√^3{9}<√^4{20}$

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, $√^3{4}>√^4{5}>√2>√^6{3}$