Simplifying Expressions Using the Properties of Exponents

SIMPLIFYING EXPRESSIONS USING THE PROPERTIES OF EXPONENTS

Properties of Exponents

x^mxⁿ = x^m+n

x^m/xⁿ = x^{m - n}

x^my^m = (xy)^m

x^m/y^m = (x/y)^m

(x^m)ⁿ = x^mn

x^-m = 1/x^m

x⁰ = 1

Solved Questions

Question 1 :

Simplify the given expression

(x²)³y⁸ / x⁵ (y⁴)³

Solution :

= (x²)³y⁸ / x⁵ (y⁴)³

= x⁶ y⁸ / x⁵ y¹²

= x y⁸ /y¹²

= x/y⁴

Question 2 :

Simplify the given expression

(x²y⁴)³ / (x⁵ y²)^-4

Solution :

= (x²y⁴)³ / (x⁵ y²)^-4

= (x²y⁴)³ (x⁵ y²)⁴

= x⁶⋅ y¹² ⋅ x²⁰ ⋅ y⁸

= x^{(6 + 20)}⋅ y^{(12 + 8)}

= x²⁶⋅ y²⁰

Question 3 :

Simplify the given expression

x¹¹(y³)² / (x³)⁵ (y²)⁴

Solution :

= x¹¹(y³)² / (x³)⁵ (y²)⁴

= x¹¹y⁶ / x¹⁵ y⁸

= 1/x⁴ y²

Question 4 :

Simplify the given expression

(x²y⁴)^-3 / (x⁵ y^-2)⁴

Solution :

= (x²y⁴)^-3 / (x⁵ y^-2)⁴

= 1/(x²y⁴)³(x⁵ y^-2)⁴

= 1/(x⁶y¹²)(x²⁰ y^-8)

= 1/(x²⁶ y⁴)

Question 5 :

Simplify the given expression

(x^-²)³y⁸ / x^-5 (y⁴)^-3

Solution :

= (x^-²)³y⁸ / x^-5 (y⁴)^-3

= x^-6y⁸ / x^-5 y^-12

= x^-6⋅ x^-5 ⋅ y⁸ ⋅ y^-12

= x^-11 ⋅ y^-4

= 1/x¹¹ ⋅ y⁴

Question 6 :

Simplify the given expression

[(x²y^-5)^-4 / (x⁵ y^-²)^-3]²

Solution :

= [(x²y^-5)^-4 / (x⁵ y^-²)^-3]²

= [(x^-8y²⁰) / (x^-15 y^-6)]²

= [x^-8⋅ x^-15y^{(20 + 6)}]²

= [x^-23y²⁶]²

= x^-46y⁵²

= y⁵²/x⁴⁶

Question 7 :

Simplify the given expression

x^-11(y³)^-2 / (x^-3)⁵ (y²)⁴

Solution :

= x^-11y^-6 / (x^-1⁵ y⁸)

= x^-11⋅ x¹⁵⋅ y^-6 ⋅ y^-8

= x⁽^{-11 + 15)}⋅ y^(-6-8)

= x⁴⋅ y^-14

= x⁴/y ¹⁴

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