SIMPLIFYING RADICALS WITH VARIABLES AND EXPONENTS

Key Concept

If the given radical is square root, write each term inside the radical as squares. 

If the given radical is cube root root, write each term inside the radical as cubes. 

If it is square root, we can get one term out of the radical for every two same terms multiplied inside the radical. 

If it is cube root, we can get one term out of the radical for every three same terms multiplied inside the radical. 

Four fourth root and more, we have to do as explained for square root and cube root above. 

Example 1 :

Simplify :

√(16u⁴v³)

Solution : 

=  √(16u⁴v³)

=  √(4² ⋅ u² ⋅ u² ⋅ v² ⋅ v)

=  (4 ⋅ u ⋅ u ⋅ v)√v

=  4u²v√v

Example 2 :

Simplify :

√(147m³n³)

Solution : 

=  √(147m³n³)

=  √(3 ⋅ 7² ⋅ m² ⋅ m ⋅ n² ⋅ n)

=  (7 ⋅ m ⋅ n)√(3mn)

=  7mn√(3mn)

Example 3 :

Simplify :

√(75x²y)

Solution : 

=  √(75x²y)

=  √(3 ⋅ 5² ⋅ x² ⋅ y)

=  (5 ⋅ x)√(3y)

=  5x√(3y)

Example 4 :

Simplify :

6√(72x²)

Solution : 

=  6√(72x²)

=  6√(2 ⋅ 6² ⋅ x²)

=  (6 ⋅ 6 ⋅ x)√2

=  36x√2

Example 5 :

Simplify :

³√(8x⁶y³)

Solution : 

=  ³√(8x⁶y³)

=  ³√(2³ ⋅ x³ ⋅ x³ ⋅ y³)

=  2 ⋅ x ⋅ x ⋅ y

=  2x²y

Example 6 :

Simplify :

³√(54m⁵n⁶)

Solution : 

=  ³√(54m⁵n⁹)

=  ³√(2 ⋅ 3³ ⋅ m³ ⋅ m² ⋅ n³ ⋅ n³ ⋅ n³)

=  (3 ⋅ m ⋅ n ⋅ n ⋅ n)³√(2 ⋅ m²)

=  3mn³√(2m²)

Example 7 :

Simplify :

⁴√(81m⁴n⁸)

Solution : 

=  ⁴√(81m⁴n⁸)

=  ⁴√(3⁴ ⋅ m⁴ ⋅ n⁴ ⋅ n⁴)

=  3 ⋅ m ⋅ n ⋅ n

=  3mn²

Example 8 :

Simplify :

⁵√(32p¹⁰q¹⁵)

Solution : 

=  ⁵√(32p¹⁰q¹⁵)

=  ⁵√(2⁵ ⋅ p⁵ ⋅ p⁵ ⋅ q⁵ ⋅ q⁵ ⋅ q⁵)

=  2 ⋅ p ⋅ p ⋅ q ⋅ q ⋅ q

=  2p²q³

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