If x and y are positive real numbers, the expression \frac{(x^{1/2} + y^{1/2})^2 - (x^{1/2… — ACT

If $x$ and $y$ are positive real numbers, the expression

\frac{(x^{1/2} + y^{1/2})^2 - (x^{1/2} - y^{1/2})^2}{(2x^{-1/4} y^{3/4})^{-2}}

can be simplified to the form $k y^p$ . What is the value of the sum $k + p$ ?

Expanding the numerator yields

4x^{1/2}y^{1/2}

. Simplifying the denominator using exponent rules yields

\frac{1}{4}x^{1/2}y^{-3/2}

. Dividing the numerator by the denominator gives

\frac{4}{1/4} \cdot x^{1/2 - 1/2} \cdot y^{1/2 - (-3/2)} = 16y^2

, which corresponds to

k = 16

and

p = 2

. The sum is

16 + 2 = 18

Expand and simplify the numerator.

4x^{1/2}y^{1/2}

Expand both squared binomials:

(x^{1/2} + y^{1/2})^2 = x + 2x^{1/2}y^{1/2} + y

and

(x^{1/2} - y^{1/2})^2 = x - 2x^{1/2}y^{1/2} + y

. Subtracting the second expression from the first yields

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

Simplify the denominator using exponent rules.

\frac{1}{4}x^{1/2}y^{-3/2}

Apply the power of a product rule

(ab)^n = a^n b^n

to distribute the exponent of

-2

(2x^{-1/4}y^{3/4})^{-2} = 2^{-2} \cdot (x^{-1/4})^{-2} \cdot (y^{3/4})^{-2}

. This simplifies to

\frac{1}{4} \cdot x^{(-1/4)(-2)} \cdot y^{(3/4)(-2)} = \frac{1}{4}x^{1/2}y^{-3/2}

Divide the simplified numerator by the simplified denominator.

16y^2

Divide the coefficients and subtract the exponents of like bases:

\frac{4x^{1/2}y^{1/2}}{\frac{1}{4}x^{1/2}y^{-3/2}} = \left(\frac{4}{1/4}\right) x^{1/2 - 1/2} y^{1/2 - (-3/2)} = 16 x^0 y^{1/2 + 3/2} = 16y^2

Identify the values of

k

and

p

and calculate their sum.

18

Comparing

16y^2

to the form

k y^p

gives

k = 16

and

p = 2

. Therefore, the sum is

k + p = 16 + 2 = 18

Properties of Exponents in Algebraic Expressions

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