**Simplifying Radical Expressions Involving Fractions :**

In this section, you will learn, how to simplify radical expressions involving fractions.

**Step 1 :**

If we have radical sign for the entire fraction, we can take separate radical for numerator and denominator.

**Step 2 :**

According to the power of radical, we can take one term out from the radical sign.

- If we have square root (√), we have to take one term out for every two same terms multiplied inside the radical.
- If we have cube root (∛), we have to take one term out for every three same terms multiplied inside the radical.
- If we have cube root (∜), we have to take one term out for every four same terms multiplied inside the radical.

**Note : **

In case we have prime number in the denominator, we have to multiply both numerator and denominator by the prime number along with radical sign.

**Step 3 :**

After taking out the terms from the radical, we have to simplify the fraction.

**Example 1 :**

Use the quotient property to write the following radical expression in simplified form.

**Solution :**

√(5/16) = √5/√16

= √5/√(4 ⋅ 4)

Index of given radical term is 2. Since its index is 2, we can factor one term for every two same terms.

= √5/4

**Example 2 :**

Use the quotient property to write the following radical expression in simplified form.

**Solution :**

√(x^{4}/25) = √x^{4}/√25

= √(x^{2} ⋅ x^{2})/√(5 ⋅ 5)

Index of given radical term is 2. Since its index is 2, we can factor one term for every two same terms.

= x^{2}/5

**Example 3 :**

Use the quotient property to write the following radical expression in simplified form.

**Solution :**

∛(4x^{2}/27) = ∛4x^{2 }/√27

= ∛(2x^{ }⋅ 2x)/√(3 ⋅ 3 ⋅ 3)

Index of given radical term is 3. Since its index is 3, we can factor one term for every three same terms.

= ∛4x^{2}/3

**Example 4 :**

Use the quotient property to write the following radical expression in simplified form.

**Solution :**

∜5x^{3}/16 = ∜5x^{3 }/ ∜16

= ∜5x^{3}/∜(2 ⋅ 2 ⋅ 2 ⋅ 2)

Index of given radical term is 4. Since its index is 4, we can factor one term for every four same terms.

= ∜5x^{3}/2

**Example 5 :**

Use the quotient property to write the following radical expression in simplified form.

**Solution :**

∜3/81a^{8} = ∜3 / ∜81a^{8}

= ∜3 / ∜(3a^{2 }⋅ 3a^{2} ⋅ 3a^{2} ⋅ 3a^{2})

Index of given radical term is 4. Since its index is 4, we can factor one term for every four same terms.

= ∜3/3a^{2}

**Example 6 :**

Use the quotient property to write the following radical expression in simplified form.

**Solution :**

√a^{6}/49 = √a^{6}/√49

= √(a^{3 }⋅ a^{3})/(7 ⋅ 7)

Index of given radical term is 2. Since its index is 2, we can factor one term for every two same terms.

= a^{3}/7

**Example 7 :**

Use the quotient property to write the following radical expression in simplified form.

**Solution :**

√5/9y^{4} = √5/√9y^{4}

= √5/(3y^{2 }⋅ 3y^{2})

= √5/3y^{2}

**Example 8 :**

Use the quotient property to write the following radical expression in simplified form.

**Solution :**

∛7/8y^{6} = ∛7 / ∛8y^{6}

= ∛7/∛(2y^{2}^{ }⋅ 2y^{2 }⋅ 2y^{2})

= ∛7/2y^{2}

After having gone through the stuff given above, we hope that the students would have understood, how to simplify radical expressions involving fractions.

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