SIMPLIFYING EXPRESSIONS WITH FRACTIONAL EXPONENTS

The following properties of exponents can be used to simplify expressions with fractional exponents. 

x^m ⋅ xⁿ  =  x^m+n

x^m ÷ xⁿ  =  x^m-n

(x^m)ⁿ  =  x^mn

(xy)^m  =  x^m ⋅ y^m

(x / y)^m  =  x^m / y^m

x^-m  =  1 / x^m

x^m/n  =  y -----> x  =  y^n/m

(x / y)^-m  =  (y / x)^m

√x  =  x^1/2

ⁿ√x  =  x^1/n

ⁿ√xⁿ  =  x

Example 1 : 

Simplify :

343^2/3

Solution : 

=  343^2/3

=  343^{1/3 ⋅ 2}

Power of a Power Property.

=  (³√343)²

=  (³√7³)²

=  7²

=  49

Example 2 : 

Simplify :

64^1/6 + 121^3/2

Solution : 

=  64^1/6 + 121^3/2 

=  64^1/6 + 121^{1/2 ⋅ 3}

Power of a Power Property.

=  64^1/6 + (121^1/2)³

Write the fractional exponents as radicals.

=  ⁶√64 + (√121)³

=  ⁶√2⁶ + (√11²)³

=  2 + (11)³

=  2 + 1331

=  1333

Example 3 : 

Simplify :

√(x⁶y⁶)

Solution : 

=  √(x⁶y⁶)

Write the square root as exponent. 

=  (x⁶y⁶)^1/2

Power of a Product Property.

=   (x⁶)^1/2 ⋅ (y⁶)^1/2

Power of a Power Property.

=   (x^6 ⋅ 1/2) ⋅ (y^6 ⋅ 1/2)

Simplify exponents. 

=   x³ ⋅ y³

=   x³y³

Example 4 : 

Simplify :

³√(a¹²b⁶)

Solution : 

=  ³√(a¹²b⁶)

Write the cube root as exponent. 

=  (a¹²b⁶)^1/3

Power of a Product Property.

=  (a¹²)^1/3 ⋅ (b⁶)^1/3

Power of a Power Property.

=  (a^12 ⋅ 1/3) ⋅ (b^6 ⋅ 1/3)

Simplify exponents. 

=  a⁴ ⋅ b²

=  a⁴b²

Example 5 : 

Simplify :

(b^1/2)² ⋅ √b²

Solution : 

=  (b^1/2)² ⋅ √b²

Power of a Power Property.

=  (b^{1/2 ⋅ 2}) ⋅ b

Simplify exponents. 

=  b ⋅ b

Product of Powers Property.

=  b^{1 + 1}

=  b²

Example 6 : 

Simplify :

[(x^1/3y^1/6)¹²] ⋅ [⁴√y⁴]

Solution : 

=  [(x^1/3y^1/6)¹²] ⋅ [⁴√y⁴]

=  (x^1/3y^1/6)¹² ⋅ y

Power of a Product Property.

=   (x^1/3)¹² ⋅ (y^1/6)¹² ⋅ y

Power of a Power Property.

=   (x^{1/3 ⋅ 12}) ⋅ (y^{1/6 ⋅ 12}) ⋅ y

=   x⁴ ⋅ y² ⋅ y

Product of powers property.

=   x⁴ ⋅ y^{2 + 1}

=   x⁴y³

Example 7 : 

Simplify :

(ab^1/2)² / ⁵√a⁵

Solution : 

=  (ab^1/2)² / ⁵√a⁵

Power of a Product Property.

=  (a)² ⋅ (b^1/2)² / a

Power of a Power Property.

=  a² ⋅ b^{1/2 ⋅ 2} / a

=  a² ⋅ b / a

Division of powers property.

=  a^{2 - 1} ⋅ b

=  ab

Example 8 : 

The approximate number of Calories C that an animal needs each day is given by C = 72m^2/3, where m is the mass of animal in kilograms. Find the number of Calories that a 27 kg dog needs each day.  

Solution : 

C  =  72m^2/3

Substitute 27 for m. 

=  72(27)^2/3

=  72(3³)^2/3

Power of a Power Property.

=  72(3^3 ⋅ 2/3)

=  72(3²)

=  72(9)

=  648

The dog needs 648 Calories per day to maintain health. 

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