Question

Simplify:

(i) (3⁵)¹¹ × (3¹⁵)⁴ – (3⁵)¹⁸ × (3⁵)⁵

(ii) (16 × 2ⁿ⁺¹ – 4 × 2ⁿ)/(16 × 2ⁿ⁺² – 2 × 2ⁿ⁺²)

(iii) (10 × 5ⁿ⁺¹ + 25 × 5ⁿ)/(3 × 5ⁿ⁺² + 10 × 5ⁿ⁺¹)

(iv) (16)⁷ ×(25)⁵× (81)³/(15)⁷ ×(24)⁵× (80)³

Answer 1

(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 (a^m)ⁿ = a^mn]

= 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 (a^m)ⁿ = a^mn]

= 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 a^m ÷ aⁿ = a^m-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 (a^m)ⁿ = a^mn]

= 5ⁿ⁺¹ (10+5)/ 5ⁿ⁺¹ (10+15)[According to the law of exponents we have a^m ÷ aⁿ = a^m-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