Simplify and write each of the following in exponential form:(i) (25)3

1.	Simplify and write each of the following in exponential form:(i) (25)3 ÷ 53(ii) (81)5 ÷ (32)5(iii) 98 × (x2)5/ (27)4 × (x3)2(iv) 32 × 78 × 136/ 212 × 913
Answer» (i) Given (25)³ ÷ 5³ = (5²)³ ÷ 5³[According to the law of exponents we have (a^m)ⁿ = a^mn] = 5⁶÷5³ [According to the law of exponents we have a^m÷ aⁿ= a^m-n] = 5^{6 – 3} = 5³ (ii) Given (81)⁵ ÷ (3²)⁵[According to the law of exponents we have (a^m)ⁿ = a^mn] = (81)⁵ ÷ 3¹⁰[81 = 3⁴] = (3⁴)⁵ ÷ 3¹⁰ [According to the law of exponents we have (a^m)ⁿ = a^mn] = 3²⁰ ÷ 3¹⁰ = 3^20-10 [According to the law of exponents we have a^m÷ aⁿ= a^m-n] = 3¹⁰ (iii) Given 9⁸ × (x²)⁵/ (27)⁴ × (x³)² = (3²)⁸ × (x²)⁵/ (3³)⁴× (x³)²[According to the law of exponents we have (a^m)ⁿ = a^mn] = 3¹⁶ × x¹⁰/3¹² × x⁶ = 3^16-12× x^10-6[According to the law of exponents we have a^m÷ aⁿ= a^m-n] = 3⁴ × x⁴ = (3x)⁴ (iv) Given (3² × 7⁸ × 13⁶)/ (21² × 91³) = (3² × 7²7⁸× 13⁶)/(21²× 13³× 7³)[According to the law of exponents we have (a^m)ⁿ = a^mn] = (21² × 7² × 13⁶)/(21²× 13³× 7³) = (7⁶× 13⁶)/(13³× 7³) = 91⁶/91³[According to the law of exponents we have a^m÷ aⁿ= a^m-n] = 91^6-3 = 91³

Simplify and write each of the following in exponential form:(i) (25)3 ÷ 53(ii) (81)5 ÷ (32)5(iii) 98 × (x2)5/ (27)4 × (x3)2(iv) 32 × 78 × 136/ 212 × 913

Answer»

(i) Given (25)³ ÷ 5³

= (5²)³ ÷ 5³[According to the law of exponents we have (a^m)ⁿ = a^mn]

= 5⁶÷5³ [According to the law of exponents we have a^m÷ aⁿ= a^m-n]

= 5^{6 – 3}

= 5³

(ii) Given (81)⁵ ÷ (3²)⁵[According to the law of exponents we have (a^m)ⁿ = a^mn]

= (81)⁵ ÷ 3¹⁰[81 = 3⁴]

= (3⁴)⁵ ÷ 3¹⁰ [According to the law of exponents we have (a^m)ⁿ = a^mn]

= 3²⁰ ÷ 3¹⁰

= 3^20-10 [According to the law of exponents we have a^m÷ aⁿ= a^m-n]

= 3¹⁰

(iii) Given 9⁸ × (x²)⁵/ (27)⁴ × (x³)²

= (3²)⁸ × (x²)⁵/ (3³)⁴× (x³)²[According to the law of exponents we have (a^m)ⁿ = a^mn]

= 3¹⁶ × x¹⁰/3¹² × x⁶

= 3^16-12× x^10-6[According to the law of exponents we have a^m÷ aⁿ= a^m-n]

= 3⁴ × x⁴

= (3x)⁴

(iv) Given (3² × 7⁸ × 13⁶)/ (21² × 91³)

= (3² × 7²7⁸× 13⁶)/(21²× 13³× 7³)[According to the law of exponents we have (a^m)ⁿ = a^mn]

= (21² × 7² × 13⁶)/(21²× 13³× 7³)

= (7⁶× 13⁶)/(13³× 7³)

= 91⁶/91³[According to the law of exponents we have a^m÷ aⁿ= a^m-n]

= 91^6-3

= 91³