SIMPLIFYING EXPRESSIONS WITH RATIONAL EXPONENTS

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

Example 1 :

Simplify : 

y^2/3 ⋅ y^7/3

Solution :

=  y^2/3 ⋅ y^7/3

Product of powers property.

=  y^{2/3 + 7/3}

Simplify exponents. 

=  y^{(2 + 7) / 3}

=  y^9/3

=  y³

Example 2 :

Simplify : 

a^3/5 ⋅ a^7/5

Solution :

=  a^3/5 ⋅ a^7/5

Product of powers property.

=  a^{3/5 + 7/5}

Simplify exponents. 

=  a^{(3 + 7) / 5}

=  a^10/5

=  a²

Example 3 : 

Simplify :

(x⁴y²)^1/2

Solution : 

=  (x⁴y²)^1/2

Power of a Product Property.

=  (x⁴)^1/2 ⋅ (y²)^1/2

Power of a Power Property.

=  (x^4 ⋅ 1/2) ⋅ (y^2 ⋅ 1/2)

Simplify exponents. 

=  (x²) ⋅ (y¹)

=  x²y

Example 4 : 

Simplify :

(a^1/2a^1/3)⁶

Solution : 

=  (a^1/2a^1/3)⁶

Power of a Product Property.

=  (a^1/2)⁶ ⋅ (a^1/3)⁶

Power of a Power Property.

=  (a^{1/2 ⋅ 6}) ⋅ (a^{1/3 ⋅ 6})

Simplify exponents. 

=  a³ ⋅ a²

Product of Powers Property.

=  a^{3 + 2}

=  a⁵

Example 5 : 

Simplify :

(2a^1/2)(3a)

Solution : 

=  (2a^1/2)(3a)

=  (3  ⋅ 2) ⋅ (a^1/2 ⋅ a)

Product of Powers Property.

=  6 ⋅ a^{1/2 + 1}

Simplify exponents. 

=  6 ⋅ a^{1/2 + 2/2}

=  6 ⋅ a^{(1 + 2)/2}

=  6a^3/2

Example 6 : 

Simplify :

√(x⁴y²)

Solution : 

=  √(x⁴y²)

Write square root as exponent. 

=  (x⁴y²)^1/2

Power of a Product Property.

=  (x⁴)^1/2 ⋅ (y²)^1/2

Power of a Power Property.

=  (x^{4 ⋅ 1/2}) ⋅ (y^{2 ⋅ 1/2})

Simplify exponents. 

=  (x²) ⋅ (y¹)

=  x²y

Example 7 : 

Simplify :

⁴√(x⁴y¹²)

Solution : 

=  ⁴√(x⁴y¹²)

Write 4^th root as exponent. 

=  (x⁴y¹²)^1/4

Power of a Product Property.

=  (x⁴)^1/4 ⋅ (y¹²)^1/4

Power of a Power Property.

=  (x^{4 ⋅ 1/4}) ⋅ (y^{12 ⋅ 1/4})

Simplify exponents. 

=  (x¹) ⋅ (y³)

=  xy³

Example 8 : 

Simplify :

(x^1/3)⁶ ⋅ ⁴√y⁴

Solution : 

=  (x^1/3)⁶ ⋅ ⁴√y⁴

Write 4^th root as exponent. 

=  (x^1/3)⁶ ⋅ (y⁴)^1/4

Power of a Power Property.

=  x^1/3 ⋅ 6 ⋅ y^{4 ⋅ 1/4}

Simplify exponents.

=  x² ⋅ y¹

=  x²y

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