Simplify (x^(1/2)+y^(1/2))^2

${(x^{\frac{1}{2}} + y^{\frac{1}{2}})}^{2}$

Step 1

Rewrite ${(x^{\frac{1}{2}} + y^{\frac{1}{2}})}^{2}$ as $(x^{\frac{1}{2}} + y^{\frac{1}{2}}) (x^{\frac{1}{2}} + y^{\frac{1}{2}})$ .

$(x^{\frac{1}{2}} + y^{\frac{1}{2}}) (x^{\frac{1}{2}} + y^{\frac{1}{2}})$

Step 2

Expand $(x^{\frac{1}{2}} + y^{\frac{1}{2}}) (x^{\frac{1}{2}} + y^{\frac{1}{2}})$ using the FOIL Method.

Apply the distributive property.

$x^{\frac{1}{2}} (x^{\frac{1}{2}} + y^{\frac{1}{2}}) + y^{\frac{1}{2}} (x^{\frac{1}{2}} + y^{\frac{1}{2}})$

Apply the distributive property.

$x^{\frac{1}{2}} x^{\frac{1}{2}} + x^{\frac{1}{2}} y^{\frac{1}{2}} + y^{\frac{1}{2}} (x^{\frac{1}{2}} + y^{\frac{1}{2}})$

Apply the distributive property.

$x^{\frac{1}{2}} x^{\frac{1}{2}} + x^{\frac{1}{2}} y^{\frac{1}{2}} + y^{\frac{1}{2}} x^{\frac{1}{2}} + y^{\frac{1}{2}} y^{\frac{1}{2}}$

Step 3

Simplify and combine like terms.

Simplify each term.

Multiply $x^{\frac{1}{2}}$ by $x^{\frac{1}{2}}$ by adding the exponents.

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x^{\frac{1}{2} + \frac{1}{2}} + x^{\frac{1}{2}} y^{\frac{1}{2}} + y^{\frac{1}{2}} x^{\frac{1}{2}} + y^{\frac{1}{2}} y^{\frac{1}{2}}$

Combine the numerators over the common denominator.

$x^{\frac{1 + 1}{2}} + x^{\frac{1}{2}} y^{\frac{1}{2}} + y^{\frac{1}{2}} x^{\frac{1}{2}} + y^{\frac{1}{2}} y^{\frac{1}{2}}$

Add $1$ and $1$ .

$x^{\frac{2}{2}} + x^{\frac{1}{2}} y^{\frac{1}{2}} + y^{\frac{1}{2}} x^{\frac{1}{2}} + y^{\frac{1}{2}} y^{\frac{1}{2}}$

Divide $2$ by $2$ .

$x^{1} + x^{\frac{1}{2}} y^{\frac{1}{2}} + y^{\frac{1}{2}} x^{\frac{1}{2}} + y^{\frac{1}{2}} y^{\frac{1}{2}}$

Simplify $x^{1}$ .

$x + x^{\frac{1}{2}} y^{\frac{1}{2}} + y^{\frac{1}{2}} x^{\frac{1}{2}} + y^{\frac{1}{2}} y^{\frac{1}{2}}$

Multiply $y^{\frac{1}{2}}$ by $y^{\frac{1}{2}}$ by adding the exponents.

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x + x^{\frac{1}{2}} y^{\frac{1}{2}} + y^{\frac{1}{2}} x^{\frac{1}{2}} + y^{\frac{1}{2} + \frac{1}{2}}$

Combine the numerators over the common denominator.

$x + x^{\frac{1}{2}} y^{\frac{1}{2}} + y^{\frac{1}{2}} x^{\frac{1}{2}} + y^{\frac{1 + 1}{2}}$

Add $1$ and $1$ .

$x + x^{\frac{1}{2}} y^{\frac{1}{2}} + y^{\frac{1}{2}} x^{\frac{1}{2}} + y^{\frac{2}{2}}$

Divide $2$ by $2$ .

$x + x^{\frac{1}{2}} y^{\frac{1}{2}} + y^{\frac{1}{2}} x^{\frac{1}{2}} + y^{1}$

Simplify $y^{1}$ .

$x + x^{\frac{1}{2}} y^{\frac{1}{2}} + y^{\frac{1}{2}} x^{\frac{1}{2}} + y$

Add $x^{\frac{1}{2}} y^{\frac{1}{2}}$ and $y^{\frac{1}{2}} x^{\frac{1}{2}}$ .

Reorder $y^{\frac{1}{2}}$ and $x^{\frac{1}{2}}$ .

$x + x^{\frac{1}{2}} y^{\frac{1}{2}} + x^{\frac{1}{2}} y^{\frac{1}{2}} + y$

Add $x^{\frac{1}{2}} y^{\frac{1}{2}}$ and $x^{\frac{1}{2}} y^{\frac{1}{2}}$ .

$x + 2 x^{\frac{1}{2}} y^{\frac{1}{2}} + y$

Step 4

Move $2 x^{\frac{1}{2}} y^{\frac{1}{2}}$ .

$x + y + 2 x^{\frac{1}{2}} y^{\frac{1}{2}}$