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 :
91/2
Solution :
= 91/2
= (3 ⋅ 3)1/2
= (32)1/2
= 32 ⋅ 1/2
= 31
= 3
Example 2 :
Evaluate :
811/2
Solution :
= 811/2
= (9 ⋅ 9)1/2
= (92)1/2
= 92 ⋅ 1/2
= 91
= 9
Example 3 :
Evaluate :
363/2
Solution :
= 363/2
= (62)3/2
= (62)3/2
= 62 ⋅ 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
= 73 ⋅ (-4/3)
= 7-4
= 1 / 74
= 1 / (7 ⋅ 7 ⋅ 7 ⋅ 7)
= 1 / 2401
Example 5 :
Evaluate :
271/6
Solution :
= 271/6
= (3 ⋅ 3 ⋅ 3)1/6
= 33 ⋅ 1/6
= 31/2
= √3
Example 6 :
Evaluate :
1005/4
Solution :
= 1005/4
= (10 ⋅ 10)5/4
= (102)5/4
= 102 ⋅ 5/4
= 105/2
= 10(2 + 1/2)
= 102 ⋅ 101/2
= 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
= 112 ⋅ 1/2 ⋅ m6 ⋅ 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
= 32 ⋅ 1/2 ⋅ n4 ⋅ 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
= 26 ⋅ 1/6 ⋅ x12 ⋅ 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
= 23 ⋅ 1/3 ⋅ x3 ⋅ 1/3 ⋅ y6 ⋅ 1/3
= 21 ⋅ x1 ⋅ y2
= 2xy2
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