SIMPLIFY EXPRESSIONS INVOLVING 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

Example 1 :

Simplify :

y^2/3 ⋅ y^7/3

Solution :

=  y^2/3 ⋅ y^7/3

=  y^{2/3 + 7/3}

=  y^{(2 + 7)/3}

=  y^9/3

=  y³

Example 2 :

Simplify :

a^3/5 ⋅ a^7/5

Solution :

=  a^3/5 ⋅ a^7/5

=  a^{3/5 + 7/5}

=  a^{(3 + 7)/5}

=  a^10/5

=  a²

Example 3 :

Simplify :

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

Solution : 

=  (2a^1/2)(3a)

=  (2 ⋅ 3)(a^1/2 ⋅ a)

=  6a^{1/2 + 1}

=  6a^{1/2 + 2/2}

=  6a^{(1 + 2)/2}

=  6a^3/2

Example 4 :

Simplify : 

(x⁴y)^1/2

Solution : 

=  (x⁴y)^1/2

=  (x⁴)^1/2(y)^1/2

=  (x^4 ⋅ 1/2)(y^1 ⋅ 1/2)

=  x²y^1/2

Example 5 :

Simplify :

(a^1/2b^1/3)²

Solution : 

=  (a^1/2b^1/3)²

=  (a^1/2)²(b^1/3)²

=  (a^1/2 ⋅ 2)(b^1/3 ⋅ 2)

=  a¹b^2/3

Example 6 :

Simplify :

(3x^-1/2 ⋅ 3x^1/2 ⋅ y^-1/3) / 3y^-7/4

Solution :

=  (3x^-1/2 ⋅ 3x^1/2 ⋅ y^-1/3) / 3y^-7/4

=  (9x^{-1/2 + 1/2} ⋅ y^-1/3) / 3y^-7/4

=  (9x⁰ ⋅ y^(-4+21)/12) / 3y^-7/4

=  (9/3) ⋅ (y^-1/3/ y^-7/4)

=  3 ⋅ y^{-1/3 + 7/4}

=  3 ⋅ y^{-4/12 + 21/12}

=  3 ⋅ y^{(-4 + 21)/12}

=  3y^17/12

Example 7 :

Simplify :

3y^1/4 / (4x^-2/3 ⋅ y^3/2 ⋅ 3y^1/2)

Solution :

=  3y^1/4 / (4x^-2/3 ⋅ y^3/2 ⋅ 3y^1/2)

=  3y^1/4 / (12x^-2/3 ⋅ y^{3/2 + 1/2})

=  3y^1/4 / (12x^-2/3 ⋅ y^{(3 + 1)/2})

=  3y^1/4 / (12x^-2/3 ⋅ y^4/2)

=  3y^1/4 / (12x^-2/3 ⋅ y²)

=  3x^2/3y^1/4 / (12y²)

=  (3/12) ⋅ (x^2/3y^1/4/ y²)

=  (1/4) ⋅ (x^2/3y^{1/4 - 2})

=  (1/4) ⋅ (x^2/3y^{1/4 - 8/4})

=  (1/4) ⋅ (x^2/3y^{(1 - 8)/4})

=  (1/4) ⋅ (x^2/3y^-7/4)

=  (1/4) ⋅ (x^2/3 / y^7/4)

 =  x^2/3 / (4y^7/4)

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