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

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

Step 1

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 2

Simplify terms.

Combine the opposite terms in $x^{\frac{1}{2}} x^{\frac{1}{2}} + x^{\frac{1}{2}} (- y^{\frac{1}{2}}) + y^{\frac{1}{2}} x^{\frac{1}{2}} + y^{\frac{1}{2}} (- y^{\frac{1}{2}})$ .

Reorder the factors in the terms $x^{\frac{1}{2}} (- y^{\frac{1}{2}})$ and $y^{\frac{1}{2}} x^{\frac{1}{2}}$ .

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

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

$x^{\frac{1}{2}} x^{\frac{1}{2}} + 0 + y^{\frac{1}{2}} (- y^{\frac{1}{2}})$

Add $x^{\frac{1}{2}} x^{\frac{1}{2}}$ and $0$ .

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

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.

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

Combine the numerators over the common denominator.

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

Add $1$ and $1$ .

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

Divide $2$ by $2$ .

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

Simplify $x^{1}$ .

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

Rewrite using the commutative property of multiplication.

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

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

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

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

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

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

Combine the numerators over the common denominator.

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

Add $1$ and $1$ .

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

Divide $2$ by $2$ .

$x - y^{1}$

Simplify $- y^{1}$ .

$x - y$

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

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