Question

Evaluate the following by using suitable identities:
(i) 304²
(ii) 509²
(iii) 992²
(iv) 799²
(v) 304 × 296
(vi) 83 × 77
(vii)109×108
(viii) 204×206

Answer 1

i) 92416

ii) 259081

iii) 984064

iv) 63840

v) 89984

vi) 6391

vii) 11772

viii) 42024

Answer 2

(i) $304^{2} =92416$

(ii) $509^{2} =259081$

(iii) $992^{2} =984064$

(iv) $799^{2} =638401$

(v) $304\times296=89984$

(vi) $83\times77=6391$

(vii) $109\times108=11772$

(viii) $204\times206=42024$

Step-by-step explanation:

Using the Identity

$(a+b)^{2} =a^{2} +2ab+b^{2}$

$(a-b)^{2} =a^{2} -2ab+b^{2}$

$(a+b)(a-b)=a^{2} -b^{2}$

$(x+a)(x+b)=x^{2} +(a+b)x+ab$

(i) $304^{2} =(300+4)^{2} =300^{2} +2\times300\times4+4^{2} =90000+2400+16=92416$

(ii) $509^{2} =(500+9)^{2} =500^{2} +2\times500\times9+9^{2} =250000+9000+81=259081$

(iii) $992^{2} =(1000-8)^{2} =1000^{2} -2\times1000\times8+8^{2} =1000000-16000+64=984064$

(iv) $799^{2} =(800-1)^{2} =800^{2} -2\times800\times1+1^{2} =640000-1600+1=638401$

(v) $304\times296=(300+4)\times(300-4)=300^{2} -4^{2} =90000-16=89984$

(vi) $83\times77=(80+3)\times(80-3)=80^{2} -3^{2} =6400-9=6391$

(vii) $109\times108=(100+9)\times(100+8)=100^{2} +100\times(9+8)+(9\times8)=10000+1700+72=11772$

(viii) $204\times206=(200+4)\times(200+6)=200^{2} +(4+6)\times200+(4\times6)=40000+2000+24=42024$