**Simplifying Radical Expressions Involving Fractions :**

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

**Step 1 :**

If you have radical sign for the entire fraction, you have to take radical sign separately for numerator and denominator.

**Step 2 :**

We have to simplify the radical term according to its power.

- If you have square root (√), you have to take one term out of the square root for every two same terms multiplied inside the radical.
- If you have cube root (
^{3}√), you have to take one term out of cube root for every three same terms multiplied inside the radical. - If you have fourth root (
^{4}√), you have to take one term out of fourth root for every four same terms multiplied inside the radical.

**Note : **

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

**Step 3 :**

After taking the terms out from radical sign, 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

√(5/16) = √5 / √(4 ⋅ 4)

Index of the given radical is 2.

Because its index is 2, we can take one term out of radical for every two same terms multiplied inside the radical sign.

√(5/16) = √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^{4}/25) = √(x^{2} ⋅ x^{2}) / √(5 ⋅ 5)

Index of the given radical is 2.

Because its index is 2, we can take one term out of radical for every two same terms multiplied inside the radical sign.

√(x^{4}/25) = x^{2 }/ 5

**Example 3 :**

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

**Solution :**

^{3}√(4x^{2}/27) = ^{3}√4x^{2 }/ ^{3}√27

^{3}√(4x^{2}/27) = ^{3}√(4x^{2}) / ^{3}√(3 ⋅ 3 ⋅ 3)

Index of the given radical is 3.

Because its index is 3, we can one term out of radical for every three same terms multiplied inside the radical sign.

^{3}√(4x^{2}/27) = ^{3}√(4x^{2}) / 3

**Example 4 :**

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

**Solution :**

^{4}√(5x^{3}/16) = ^{4}√(5x^{3}) / ^{4}√16

^{4}√(5x^{3}/16) = ^{4}√5x^{3 }/ ^{4}√(2 ⋅ 2 ⋅ 2 ⋅ 2)

Index of the given radical is 4.

Because its index is 4, we can take one term out of the radical for every four same terms multiplied inside the radical sign.

^{4}√(5x^{3}/16) = ^{4}√5x^{3 }/ 2

**Example 5 :**

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

**Solution :**

^{4}√(3/81a^{8}) = ^{4}√3 / ^{4}√(81a^{8})

^{4}√(3/81a^{8}) = ^{4}√3 / ^{4}√(3a^{2 }⋅ 3a^{2} ⋅ 3a^{2} ⋅ 3a^{2})

Index of the given radical is 4.

Because its index is 4, we can take one term out of the radical for every four same terms multiplied inside the radical sign.

^{4}√(3/81a^{8}) = ^{4}√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^{6}/49) = √(a^{3 }⋅ a^{3})/(7 ⋅ 7)

Index of the given radical is 2.

Because its index is 2, we can take one term out of the radical for every two same terms multiplied inside the radical sign.

√(a^{6}/49) = 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}

√(5/9y^{4}) ^{ }= √5 / (3y^{2 }⋅ 3y^{2})

Index of the given radical is 2.

Because its index is 2, we can take one term out of the radical for every two same terms multiplied inside the radical sign.

√(5/9y^{4})^{ }= √5 / 3y^{2}

**Example 8 :**

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

**Solution :**

^{3}√(7/8y^{6}) = ^{3}√7 / ^{3}√(8y^{6})

^{3}√(7/8y^{6}) = ^{3}√7 / ^{3}√(2y^{2}^{ }⋅ 2y^{2 }⋅ 2y^{2})

Index of the given radical is 3.

Because its index is 3, we can take one term out of the radical for every three same terms multiplied inside the radical sign.

^{3}√(7/8y^{6}) = ^{3}√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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