**Division with rational exponents :**

Here we are going to see how to divide rational exponents.

**Division property rational exponents :**

If we have same base for both numerator and denominator, we have to put only one base and subtract the powers.

**x ^{a} ÷ x^{b } = x^{(a-b)}**

**If we have same power for both numerator and denominator, we have to distribute the power for both numerator and denominator.**

**(a/b) ^{m}^{ } = a^{m}/b^{m}**

**Let us look into some example problems based on the above concept.**

**Example 1 :**

Simplify the following rational exponents

10^{9} ÷ 10^{6}

**Solution :**

= 10^{9}/10^{6}

= 10^{(9-6)}

= 10^{3 }==> 10 x 10 x 10 ==> 1000

**Example 2 :**

Simplify the following rational exponents

3a^{1/2} ÷ a^{1/3}

**Solution :**

= 3 a^{1/2} ÷ a^{1/3}

= 3 a^{1/2} ⋅ a^{-1/3}

= 3 a^{(1/2) - (}^{1/3)}

= 3 a^{(3 - 2)/6}

= 3 a^{1/6}

**Example 3 :**

Simplify the following rational exponents

3y^{1/4} ÷ 6y^{1/2}

**Solution :**

= 3y^{1/4} ÷ 6y^{1/2}

= 3y^{1/4} / 6y^{1/2}

= (3/6) y ^{(1/4)-(}^{1/2)}

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

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

= 1/2y^{1/4}

**Example 4 :**

Simplify the following rational exponents

(m^{5/3} ÷ m)^{2}

**Solution :**

= (m^{5/3} / m)^{2}

= (m^{5/3}⋅ m^{-1})^{2}

= (m^{(5/3) - 1})^{2}

= (m^{(5-3)/3})^{2}

= (m^{2/3})^{2}

= m^{4/3}

**Example 5 :**

Simplify the following rational exponents

**Solution :**

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

First let us combine the x and y terms

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

= [(x^{(1/2) + (7}^{/4) }y(^{-2 - 1})]^{4}

= [(x^{(2 + 7)/4}^{ }y^{-3}]^{4}

= [(x^{9/4}^{ }y^{-3}]^{4}

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

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

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

**Example 6 :**

Simplify the following rational exponents

**Solution :**

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

Let us distribute the powers for both numerator and denominator separately.

= (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}y^{3} x^{1/4 }y^{1/6}

= x^{9/2}⋅ x^{1/4}y^{3}^{ }⋅ y^{1/6}

= x^{(9/2) + (}^{1/4) }y^{(3 + (}^{1/6))}

= x^{(18 + 1)/4}^{ }y^{(18 + }^{1)/6)}

= x^{19/4}^{ }y^{19}^{/6}

**Example 7 :**

Simplify the following rational exponents

**Solution :**

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

Let us distribute the powers for both numerator

= [(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 }y^{-5/2}^{ }⋅ x^{-2 }^{ }⋅ y^{-1/2})]

= x^{5/8 }^{ }⋅ x^{-2 }y^{-5/2}^{ }⋅ y^{-1/2}

= x^{((5/8)}^{-2) }y^{(-5/2)}^{-(1/2)}

= x^{((5 - 16)/8}^{ }y^{(-5 - 1)/2}

= x^{ -11/8}^{ }y^{-6/2}

= x^{ -11/8}^{ }y^{-3}

In order to make the power as positive, we have to write it in the denominator.

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

- Properties of radicals
- Simplifying radical expressions worksheets
- Square roots
- Ordering square roots from least to greatest
- Operations with radicals
- How to simplify radical expressions

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