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

Answer: (b)

The orders of the given surds are 3, 2, 4 and 6.

Their LCM = 12

Now we convert each surd into a surd of order 12.

$√^3{9} = (9)^{1/3} =(9)^{4/12} = (9^4)^{1/12}=√^12{6561}$

Similarly,

$√3 =√^12{3^6}=√^12{729}$

$√^4{16} =√^12{16^3}=√^12{4096}$

$√^6{80} =√^12{80^2}=√^12{6400}$

Clearly,

$√^12{729}<√^12{4096}<√^12{6400}<√^12{6561}$

∴ $√^3{9}$ is the greatest number.

Answer: (c)

$√2, √^3{3}, √^4{4}$

LCM of power of surds = 12

$√2=(2)^{1/2}=(2^6)^{1/12}$

=$√^12{2^6}=√^12{64}$

$√^3{3}=√^12{3^4}=√^12{81}$

$√^4{4}=√^12{4^3}=√^12{64}$

Since 81 is the largest, hence,

$√^3{3}$ is the largest number.

Answer: (c)

$√7-√5, √5-√3, √9-√7, √{11}-√9$

$1/{√7-√5}={√7+√5}/{(√7-√5)(√7+√5)}$

=${√7+√5}/{7-5}={√7+√5}/2$,

$1/{√5-√3}={√5+√3}/{(√5-√3)(√5+√3)}$

=${√5+√3}/{5-3}={√5+√3}/2$

Similarly,

$1/{√9-√7}={√9+√7}/2$

$1/{√{11}-√9}={√{11}+√9}/2$

Clearly,${√5+√3}/2$ is the smallest.

$1/{√5-√3}$ is the smallest.

∴ $√5-√3$ is the greatest.

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)

$√2, √^3{9}, √^4{16}, √^5{32}$

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

$√^5{32}=(32)^{1/5}=(2^5)^{1/5}=2$

$√^3{9}>2,√2<2$

Answer: (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.)