variables with fractional exponents

Question

In Algebra 2, I am given the following problem:

 

(x^3/4 + y^-1/2)²

 

I am told to expand and simplify. How do I solve this?

Answer 1

Expand the expression

 

 

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

 

x^6/4 + 2x^3/4y^-1/2 + y^-1 =

 

x^3/2 + 2x^3/4y^-1/2 + y^-1

 

 

Now we can put them in radical form.

 

√(x³) + 2  ⁴√(x³) √(1/y) + (1/y) =

 

√(x²) √(x) + 2  ⁴ √(x³)  *  1/√(y) + 1/y =

 

x√(x) + 2  ⁴√(x³) / √(y) + 1/y

 

 

Now to finish, use the LCD and rationalize the denominator.

Answer 2

(x^3/4 + y^-1/2)²      Either FOIL or the general form can be used. Worry about the exponents at the end.

 

(x^3/4 + y^-1/2)(x^3/4 + y^-1/2)

x^3/4x^3/4 + x^3/4y^-1/2 + x^3/4y^-1/2 + y^-1/2y^-1/2   Simplify the exponents.

x^6/4 + x^3/4y^-1/2 + x^3/4y^-1/2 + y^-1                       Add like terms

x^6/4 +2x^3/4y^-1/2 + y^-1                                            From here you decide whether to write only positive exponents.

 

OK?

 

Answer 3

(A + B)² = A² + 2AB + B²

 

So, (x^3/4 + y^-1/2)² = (x^3/4)² + 2(x^3/4)(y^-1/2) + (y^-1/2)²

 

                                  = x^3/2 + 2x^3/4y^-1/2 + y^-1