SIMPLIFYING RATIONAL EXPONENTS

If we have power raised to another power, we have to multiply the powers.

If we have numerical values inside the parenthesis,

• Decompose it as much as possible
• Write it in exponential form
• Then multiply the powers for further simplification.

Simplify the following without using calculator.

Example 1 :

(-32)^2/5

Solution :

= (-32)^2/5

Decomposing 32, we get

-32 = (-2) ⋅ (-2) ⋅ (-2) ⋅ (-2) ⋅ (-2)

-32 = (-2)⁵

(-32)^2/5  = [(-2)⁵]^2/5

Since we have power raised to another power, we have to multiply the powers.

= (-2)^5 ⋅ 2/5

= (-2)²

(-32)^2/5 = 4

Example 2 :

(-8)^4/3

Solution :

= (-8)^4/3

Decomposing -8, we get

-8 = (-2) ⋅ (-2) ⋅ (-2)

-8 = (-2)³

(-8)^4/3  = [(-2)³]^4/3

= (-2)³

= -8

Example 3 :

(8/27)^4/3

Solution :

= (8/27)^4/3

Decomposing 8 and 27, we get

(8/27)^4/3 = [(2/3)³]^4/3

= (2/3) ^{3 ⋅ (4/3)}

= (2/3)⁴

= 16/81

Example 4 :

(125)^1/3

Solution :

= (125)^1/3

Decomposing 125, we get

125 = 5 ⋅ 5 ⋅ 5 ==> 5³

= (5³)^1/3

= 5^{3 ⋅ (1/3)}

= 5

Example 5 :

(8x¹⁵)^-1/3

Solution :

= (8x¹⁵)^-1/3

Decomposing 8, we get

8 = 2 ⋅ 2 ⋅ 2 ==> 2³

= (2³ (x⁵)³)^-1/3

= (2x⁵)^{3 ⋅ (-1/3)}

= (2x⁵)^-1

=1/(2x⁵)

Example 6 :

(h⁶p⁹/1000m³)^-2/3

Solution :

Example 7 :

(64)^1/6

Solution :

= (64)^1/6

Decomposing 64, we get

64 = 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2

64 = 2⁶

(64)^1/6 = (2⁶)^1/6

= 2 ^{(6 ⋅ 1/6)}

= 2

Example 8 :

(256)^3/4

Solution :

= (256)^3/4

Decomposing 256, we get

256 = 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2

256 = 4⁴

(256)^3/4 = (4⁴)^3/4

= 4³

= 64