(i) Given (35)11 × (315)4 – (35)18 × (35)5
= (3)55 × (3)60 – (3)90 × (3)25[According to the law of exponents we have (am)n = amn]
= 3 55+60 – 390+25
= 3115 – 3115
= 0
(ii) Given (16 × 2n+1 – 4 × 2n)/(16 × 2n+2 – 2 × 2n+2)
= (24 × 2(n+1) -22 × 2n)/(24 × 2(n+2) -22+1 × 22) [According to the law of exponents we have (am)n = amn]
= 22 × 2(n+3-2n)/)22× 2(n+4-2n+1)
= 2n × 23 – 2n/ 2n × 24 – 2n × 2
= 2n(23 – 1)/ 2n(24 – 1) [According to the law of exponents we have am ÷ an = am-n]
= 8 -1 /16 -2
= 7/14
= (1/2)
(iii) Given (10 × 5n+1 + 25 × 5n)/(3 × 5n+2 + 10 × 5n+1)
= (10 × 5n+1 + 52 × 5n)/(3 × 5n+2 + (2 × 5) × 5n+1)
= (10 × 5n+1 + 5 × 5n+1)/(3 × 5n+2 + (2 × 5) × 5n+1) [According to the law of exponents we have (am)n = amn]
= 5n+1 (10+5)/ 5n+1 (10+15)[According to the law of exponents we have am ÷ an = am-n]
= 15/25
= (3/5)
(iv) Given (16)7 ×(25)5× (81)3/(15)7 ×(24)5× (80)3
= (16)7 ×(52)5× (34)3/(3 × 5 )7 ×(3 × 8)5× (16 × 5)3
= (16)7 ×(52)5× (34)3/37 × 57 × 35 × 85× 163× 53
= (16)7/ 85 × 16 3
= (16)4/85
= (2 × 8)4/85
= 24/8
= (16/8)
= 2