SIMPLIFYING RADICAL EXPRESSIONS INVOLVING FRACTIONS

Simplifying Radical Expressions Involving Fractions :

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

Quotient Property of Radicals

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.

Simplifying Radical Expressions Involving Fractions - Examples

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 :

√(x⁴/25) = √x⁴/√25

= √(x² ⋅ x²)/√(5 ⋅ 5)

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

= x²/5

Example 3 :

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

Solution :

∛(4x²/27) = ∛4x²/√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.

= ∛4x²/3

Example 4 :

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

Solution :

∜5x³/16 = ∜5x³/ ∜16

= ∜5x³/∜(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.

= ∜5x³/2

Example 5 :

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

Solution :

∜3/81a⁸ = ∜3 / ∜81a⁸

= ∜3 / ∜(3a²⋅ 3a² ⋅ 3a² ⋅ 3a²)

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

= ∜3/3a²

Example 6 :

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

Solution :

√a⁶/49 = √a⁶/√49

= √(a³⋅ a³)/(7 ⋅ 7)

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

= a³/7

Example 7 :

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

Solution :

√5/9y⁴ = √5/√9y⁴

= √5/(3y²⋅ 3y²)

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

= √5/3y²

Example 8 :

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

Solution :

∛7/8y⁶ = ∛7 / ∛8y⁶

= ∛7/∛(2y²⋅ 2y²⋅ 2y²)

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

= ∛7/2y²

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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