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

x^{m} ⋅ x^{n} = x^{m+n}

x^{m} ÷ x^{n} = 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}

= 10^{2}

= 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^{1/2}y^{-2}) / (yx^{-7/4})]^{4}

**Solution :**

= [(x^{1/2}y^{-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}

= x^{9}^{ }⋅ y^{-12}

= x^{9} / y^{12}

**Example 6 :**

Simplify :

(x^{3}y^{2})^{3/2} / (x^{-1 }y^{-2/3})^{1/4}

**Solution :**

= (x^{3}y^{2})^{3/2} / (x^{-1 }y^{-2/3})^{1/4}

= (x^{3})^{3/2}(y^{2})^{3/2} / (x^{-1})^{1/4 }(y^{-2/3})^{1/4}

= x^{9/2}y^{3} / 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^{-1/2}y^{2})^{-5/4} / (x^{2 }y^{1/2})

**Solution :**

= (x^{-1/2}y^{2})^{-5/4} / (x^{2 }y^{1/2})

= (x^{-1/2})^{-5/4}(y^{2})^{-5/4} / (x^{2 }y^{1/2})

= x^{5/8 }y^{-5/2} / (x^{2 }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^{-11/8}y^{-3}

= 1 / (x^{11/8}y^{3})

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