EVALUATE INTEGERS RAISED TO RATIONAL EXPONENTS

The following steps will be useful to evaluate integers with rational exponents. 

Step 1 :

Write the base in exponential notation

Step 2 :

Multiply the two exponents simplify. 

Example 1 :

Evaluate :

9^1/2

Solution :

=  9^1/2

=  (3 ⋅ 3)^1/2

=  (3²)^1/2

=  3^2 ⋅ 1/2

=  3¹

  =  3

Example 2 :

Evaluate : 

81^1/2

Solution :

=  81^1/2

=  (9 ⋅ 9)^1/2

=  (9²)^1/2

=  9^{2 ⋅ 1/2}

=  9¹

=  9

Example 3 :

Evaluate :

36^3/2

Solution :

=  36^3/2

=  (6²)^3/2

=  (6²)^3/2 

=  6^{2 ⋅ 3/2}

=  6³

=  6 ⋅ 6 ⋅ 6

=  216

Example 4 :

Evaluate :

343^-4/3

Solution :

=  343^-4/3

=  (7 ⋅ 7 ⋅ 7)^-4/3

=  (7³)^-4/3

=  7^{3 ⋅ (-4/3)}

=  7^-4

  =  1 / 7⁴

=  1 / (7 ⋅ 7 ⋅ 7 ⋅ 7)

=  1 / 2401

Example 5 :

Evaluate : 

27^1/6

Solution :

=  27^1/6

=  (3 ⋅ 3 ⋅ 3)^1/6

=  3^{3 ⋅ 1/6}

=  3^1/2

=  √3

Example 6 :

Evaluate :

100^5/4

Solution :

=  100^5/4

=  (10 ⋅ 10)^5/4

=  (10²)^5/4 

=  10^{2 ⋅ 5/4}

=  10^5/2

=  10^{(2 + 1/2)}

=  10² ⋅ 10^1/2

=  100 ⋅ √10

=  100√10

Example 7 :

Evaluate : 

(121m⁶)^1/2

Solution :

=  (121m⁶)^1/2

=  (11 ⋅ 11 ⋅ m⁶)^1/2

=  (11² ⋅ m⁶)^1/2

=  (11²)^1/2 ⋅ (m⁶)^1/2

=  11^{2 ⋅ 1/2} ⋅ m^{6 ⋅ 1/2}

=  11¹ ⋅ m³

=  11m³

Example 8 :

Evaluate : 

(9n⁴)^1/2

Solution :

=  (9n⁴)^1/2

=  (3² ⋅ n⁴)^1/2

=  (3²)^1/2 ⋅ (n⁴)^1/2

=  3^{2 ⋅ 1/2} ⋅ n^{4 ⋅ 1/2}

=  3¹ ⋅ n²

=  3 ⋅ n²

=  3n²

Example 9 :

Evaluate : 

(64x¹²)^1/6

Solution :

=  (64x¹²)^1/6

=  (2⁶ ⋅ x¹²)^1/6

=  (2⁶)^1/6 ⋅ (x¹²)^1/6

=  2^{6 ⋅ 1/6} ⋅ x^{12 ⋅ 1/6}

=  2 ⋅ x²

=  2x²

Example 10 :

Evaluate : 

(8x³y⁶)^1/3

Solution :

=  (8x³y⁶)^1/3

=  (2³ ⋅ x³ ⋅ y⁶)^1/3

=  (2³)^1/3 ⋅ (x³)^1/3 ⋅ (y⁶)^1/3 

=  2^{3 ⋅ 1/3} ⋅ x^{3 ⋅ 1/3} ⋅ y^{6 ⋅ 1/3} 

=  2¹ ⋅ x¹ ⋅ y²

=  2xy²

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