Evaluate each of the following

Solution:

Given,

(a) $(103)^{3}$

we know that $(a+b)^{3}=a^{3}+b^{3}+3 a b(a+b)$

$\Rightarrow(103)^{3}$ can be written as $(100+3)^{3}$

Here, a = 100 and b = 3

$(103)^{3}=(100+3)^{3}$

$=(100)^{3}+(3)^{3}+3(100)(3)(100+3)$

= 1000000 + 27 + (900*103)

= 1000000 + 27 + 92700

= 1092727

The value of $(103)^{3}=1092727$

(b) $(98)^{3}$

we know that $(a-b)^{3}=a^{3}-b^{3}-3 a b(a-b)$

$\Rightarrow(98)^{3}$ can be written as $(100-2)^{3}$

Here, a = 100 and b = 2

$(98)^{3}=(100-2)^{3}$

$=(100)^{3}-(2)^{3}-3(100)(2)(100-2)$

= 1000000 - 8 - (600*102)

= 1000000 – 8 – 58800

= 941192

The value of $(98)^{3}=941192$

(c) $(9.9)^{3}$

we know that $(a-b)^{3}=a^{3}-b^{3}-3 a b(a-b)$

$\Rightarrow(9.9)^{3}$ can be written as $(10-0.1)^{3}$

Here, a = 10 and b = 0.1

$(9.9)^{3}=(10-0.1)^{3}$

$=(10)^{3}-(0.1)^{3}-3(10)(0.1)(10-0.1)$

= 1000 – 0.001 - (3*9.9)

= 1000 – 0.001 – 29.7

= 1000 – 29.701

= 970.299

The value of $(9.9)^{3}=970.299$

(d) $(10.4)^{3}$

we know that $(a+b)^{3}=a^{3}+b^{3}+3 a b(a+b)$

$\Rightarrow(10.4)^{3}$ can be written as $(10+0.4)^{3}$

Here, a = 10 and b = 0.4

$(10.4)^{3}=(10+0.4)^{3}$

$=(10)^{3}+(0.4)^{3}+3(10)(0.4)(10+0.4)$

= 1000 + 0.064 + (12*10.4)

= 1000 + 0.064 + 124.8

= 1000 + 124.864

= 1124.864

The value of $(10.4)^{3}=1124.864$

(e) $(598)^{3}$

we know that $(a-b)^{3}=a^{3}-b^{3}-3 a b(a-b)$

$\Rightarrow(598)^{3}$ can be written as $(600-2)^{3}$

Here, a = 600 and b = 2

$(598)^{3}=(600-2)^{3}$

$=(600)^{3}-(2)^{3}-3(600)(2)(600-2)$

= 216000000 - 8 - (3600*598)

= 216000000 - 8 - 2152800

= 216000000 - 2152808

= 213847192

The value of $(598)^{3}=213847192$

(f) (99) ${ }^{3}$

we know that $(a-b)^{3}=a^{3}-b^{3}-3 a b(a-b)$

$\Rightarrow(99)^{3}$ can be written as $(100-1)^{3}$

Here , a = 100 and b = 1

$(99)^{3}=(100-1)^{3}$C

$=(100)^{3}-(1)^{3}-3(100)(1)(100-1)$

= 1000000 - 1 - (300*99)

= 1000000 - 1 - 29700

= 1000000 - 29701

= 970299

The value of $(99)^{3}=970299$