Evaluate each of the following

Solution:

(a) Given,

$111^{3}-89^{3}$

the above equation can be written as $(100+11)^{3}-(100-11)^{3}$

we know that, $(a+b)^{3}-(a-b)^{3}=2\left[b^{3}+3 a b^{2}\right]$

here, a = 100 b = 11

$(100+11)^{3}-(100-11)^{3}=2\left[11^{3}+3(100)^{2}(11)\right]$

= 2[1331 + 330000]

= 2[331331]

= 662662

The value of $111^{3}-89^{3}=662662$

(b) $46^{3}+34^{3}$

the above equation can be written as $(40+6)^{3}+(40-6)^{3}$

we know that, $(a+b)^{3}+(a-b)^{3}=2\left[a^{3}+3 a b^{2}\right]$

here, a= 40 , b = 4

$(40+6)^{3}+(40-6)^{3}=2\left[40^{3}+3(6)^{2}(40)\right]$

= 2[64000 + 4320]

= 2[68320]

= 1366340

The value of $46^{3}+34^{3}=1366340$

(c) $104^{3}+96^{3}$

the above equation can be written as $(100+4)^{3}+(100-4)^{3}$

we know that, $(a+b)^{3}+(a-b)^{3}=2\left[a^{3}+3 a b^{2}\right]$

here, a= 100 b = 4

$(100+4)^{3}-(100-4)^{3}=2\left[100^{3}+3(4)^{2}(100)\right]$

= 2[1000000 + 4800]

= 2[1004800]

= 2009600

The value of $104^{3}+96^{3}=2009600$

(d) $93^{3}-107^{3}$

the above equation can be written as $(100-7)^{3}-(100+7)^{3}$

we know that, $(a-b)^{3}-(a+b)^{3}=-2\left[b^{3}+3 b a^{2}\right]$

here, a = 93,  b = 107

$(100-7)^{3}-(100+7)^{3}=-2\left[7^{3}+3(100)^{2}(7)\right]$

= - 2[343 + 210000]

= - 2[210343]

= - 420686

The value of $93^{3}-107^{3}=-420686$