MULTIPLICATION WITH RATIONAL EXPONENTS

Example 1 :

Evaluate :

3x^1/2x^2/3

Solution :

=  3x^1/2 ⋅ x^2/3

=  3x^{1/2 + 2/3}

=   3x^{3/6 + 4/6}

=   3x^7/6

Example 2 : 

Evaluate : 

(³√x² ⋅ ⁶√x⁴)^-3

Solution : 

=  (³√x² ⋅ ⁶√x⁴)^-3

=  [(x²)^1/3 ⋅ (x⁴)^1/6]^-3

=  [x^2/3 ⋅ x^2/3]^-3

=  (x^4/3)^-3

=  x^-4

=  1/x⁴

Example 3 : 

Evaluate : 

(³√x⁴ ⋅ √x⁵) / √(25x¹⁶)

Solution : 

=  (³√x⁴ ⋅ √x⁵) / √(25x¹⁶)

=  [(x⁴)^1/3 ⋅ (x⁵)^1/2] / (25x¹⁶)^1/2

=  (x^4/3 ⋅ x^5/2) / (5x⁸)

=  (x^{4/3 + 5/2 - 8}) / 5

=  (x^{8/6 + 15/6 - 48/6}) / 5

=  (x^-25/6) / 5

=  1 / (5x^25/6)

Example 4 :

Evaluate :

√(8x³)√(18x)

Solution :

=  √(8x³ ⋅ 18x)

=  √(144x⁴)

=  (144x⁴)^1/2

=  12x²

Example 5 : 

Evaluate : 

(216^1/3 ⋅ 6^1/3)^3/2

Solution :

=  [(6³)^1/3 ⋅ 6^1/3]^3/2

=  (6 ⋅ 6^1/3)^3/2

  =  (6^{1 + 1/3})^3/2

=  (6^{3/3 + 1/3})^3/2

=  (6^4/3)^3/2

=  6²

=  36

Example 6 :

Evaluate :

³√(8x³) ⋅ ⁴√(625x⁸)

Solution :

=  (8x³)^1/3 ⋅ (625x⁸)^1/4

=  (8x³)^1/3 ⋅ (625x⁸)^1/4

=  (2³)^1/3(x³)^1/3 ⋅ (5⁴)^1/4(x⁸)^1/4

=  2x ⋅ 5x²

=  10x^{1 + 2}

=  10x³

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