## DIVISION WITH RATIONAL EXPONENTS

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

xm ⋅ xn  =  xm+n

xm ÷ xn  =  xm-n

(xm)n  =  xmn

(xy)m  =  xm ⋅ ym

(x / y)m  =  xm / ym

x-m  =  1 / xm

xm/n  =  y -----> x  =  yn/m

(x / y)-m  =  (y / x)m

Example 1 :

Evaluate :

105/2 ÷ 101/2

Solution :

=  105/2 ÷ 101/2

=  10(5/2 - 1/2)

=  10(5 - 1)/2

=  104/2

=  102

=  10 ⋅ 10

=  100

Example 2 :

Simplify :

3a1/2 ÷ a1/3

Solution :

=  3a1/2 ÷ a1/3

=  3a(1/2 - 1/3)

=  3a(3/6 - 2/6)

=  3a(3 - 2)/6

=  3a1/6

Example 3 :

Simplify :

3y1/4 ÷ 6y1/2

Solution :

=  3y1/4 ÷ 6y1/2

=  3y1/4 / 6y1/2

=  (3/6) ⋅ (y1/4 / 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 / (2y1/4)

Example 4 :

Simplify :

(m5/3 ÷ m)2

Solution :

=  (m5/3 / m)2

=  (m(5/3 - 1))2

=  (m(5/3 - 3/3))2

=  (m(5-3)/3)2

=  (m2/3)2

=  m2/3 ⋅ 2

=  m4/3

Example 5 :

Simplify :

[(x1/2y-2) / (yx-7/4)]4

Solution :

=  [(x1/2y-2) / (yx-7/4)]4

(x1/2 + 7/4y-2 -1)4

=  (x2/4 + 7/4y-3)4

=  [(x(2 + 7)/4 y-3]4

=  [(x9/4 y-3]4

=  (x9/4)4(y-3)4

=  x9/4 ⋅ 4 ⋅ y-⋅ 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

=  x9/2y3 / x-1/4 y-1/6

=  x9/2 + 1/4y3 + 1/6

=  x18/4 + 1/4y18/6 + 1/6

=  x(18 + 1)/4y(18 + 1)/6

=  x19/4 y19/6

Example 7 :

Simplify :

(x-1/2y2)-5/4 / (xy1/2)

Solution :

=  (x-1/2y2)-5/4 / (xy1/2)

(x-1/2)-5/4(y2)-5/4 / (xy1/2)

=  x5/8 y-5/2 / (xy1/2)

=  x5/8 - 2y-5/2 - 1/2

=  x5/8 - 16/8y(-5 - 1)/2

=  x(5 - 16)/8y-6/2

=  x-11/8y-3

=  1 / (x11/8y3)

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