DIVISION WITH RATIONAL EXPONENTS

The following properties of exponents can be used to do division with rational exponents. 

x^m ⋅ xⁿ  =  x^m+n

x^m ÷ xⁿ  =  x^m-n

(x^m)ⁿ  =  x^mn

(xy)^m  =  x^m ⋅ y^m

(x / y)^m  =  x^m / y^m

x^-m  =  1 / x^m

x^m/n  =  y -----> x  =  y^n/m

(x / y)^-m  =  (y / x)^m

Example 1 :

Evaluate :

10^5/2 ÷ 10^1/2

Solution : 

=  10^5/2 ÷ 10^1/2

=  10^{(5/2 - 1/2)}

=  10^{(5 - 1)/2}

=  10^4/2

=  10²

=  10 ⋅ 10

=  100

Example 2 :

Simplify :

3a^1/2 ÷ a^1/3

Solution : 

 =  3a^1/2 ÷ a^1/3

 =  3a^{(1/2 - 1/3)}

=  3a^{(3/6 - 2/6)}

=  3a^{(3 - 2)/6}

=  3a^1/6

Example 3 :

Simplify :

3y^1/4 ÷ 6y^1/2

Solution : 

=  3y^1/4 ÷ 6y^1/2

=  3y^1/4 / 6y^1/2

=  (3/6) ⋅ (y^1/4 / y ^1/2)

=  (1/2) ⋅ (y^{(1/4 - 1/2)}

=  (1/2) ⋅ (y^{(1/4 - 2/4)}

=  (1/2) ⋅ (y^{(1 - 2)/4}

=  (1/2) ⋅ y^-1/4

=  1 / (2y^1/4)

Example 4 :

Simplify :

(m^5/3 ÷ m)²

Solution : 

=  (m^5/3 / m)²

=  (m^{(5/3 - 1)})²

=  (m^{(5/3 - 3/3)})²

=  (m^(5-3)/3)²

=  (m^2/3)²

=  m^{2/3 ⋅ 2}

=  m^4/3

Example 5 :

Simplify :

[(x^1/2y^-2) / (yx^-7/4)]⁴

Solution : 

=  [(x^1/2y^-2) / (yx^-7/4)]⁴

=  (x^{1/2 + 7/4}y^{-2 -1})⁴

=  (x^{2/4 + 7/4}y^-3)⁴

=  [(x^{(2 + 7)/4} y^-3]⁴

=  [(x^9/4 y^-3]⁴

=  (x^9/4)⁴(y^-3)⁴

=  x^{9/4 ⋅ 4} ⋅ y^{-3 ⋅ 4}

=  x⁹ ⋅ y^-12

=  x⁹ / y¹²

Example 6 :

Simplify :

(x³y²)^3/2 / (x^-1 y^-2/3)^1/4

Solution : 

=  (x³y²)^3/2 / (x^-1 y^-2/3)^1/4

=  (x³)^3/2(y²)^3/2 / (x^-1)^1/4 (y^-2/3)^1/4

=  x^9/2y³ / x^-1/4 y^-1/6

=  x^{9/2 + 1/4}y^{3 + 1/6}

=  x^{18/4 + 1/4}y^{18/6 + 1/6}

=  x^{(18 + 1)/4}y^{(18 + 1)/6}

=  x^19/4 y^19/6

Example 7 :

Simplify : 

(x^-1/2y²)^-5/4 / (x² y^1/2)

Solution : 

=  (x^-1/2y²)^-5/4 / (x² y^1/2)

=  (x^-1/2)^-5/4(y²)^-5/4 / (x² y^1/2)

=  x^5/8 y^-5/2 / (x² y^1/2)

=  x^{5/8 - 2}y^{-5/2 - 1/2}

=  x^{5/8 - 16/8}y^{(-5 - 1)/2}

=  x^{(5 - 16)/8}y^-6/2

=  x^-11/8y^-3

  =  1 / (x^11/8y³)

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