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 $x^{1}$ .

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

Rewrite using the commutative property of multiplication.

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

Rewrite using the commutative property of multiplication.

$x - x^{\frac{1}{2}} y^{\frac{1}{2}} - y^{\frac{1}{2}} x^{\frac{1}{2}} - 1 \cdot - 1 y^{\frac{1}{2}} y^{\frac{1}{2}}$

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

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

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

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

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

Combine the numerators over the common denominator.

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

Add $1$ and $1$ .

$x - x^{\frac{1}{2}} y^{\frac{1}{2}} - y^{\frac{1}{2}} x^{\frac{1}{2}} - 1 \cdot - 1 y^{\frac{2}{2}}$

Divide $2$ by $2$ .

$x - x^{\frac{1}{2}} y^{\frac{1}{2}} - y^{\frac{1}{2}} x^{\frac{1}{2}} - 1 \cdot - 1 y^{1}$

Simplify $- 1 \cdot - 1 y^{1}$ .

$x - x^{\frac{1}{2}} y^{\frac{1}{2}} - y^{\frac{1}{2}} x^{\frac{1}{2}} - 1 \cdot - 1 y$

Multiply $- 1$ by $- 1$ .

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

Multiply $y$ by $1$ .

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

Subtract $y^{\frac{1}{2}} x^{\frac{1}{2}}$ from $- x^{\frac{1}{2}} y^{\frac{1}{2}}$ .

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

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

Subtract $x^{\frac{1}{2}} y^{\frac{1}{2}}$ from $- x^{\frac{1}{2}} y^{\frac{1}{2}}$ .

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

Step 4

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

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

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

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