Simplify ( square root of u+ square root of v)( square root of u- square root of v)

$(\sqrt{u} + \sqrt{v}) (\sqrt{u} - \sqrt{v})$

Step 1

Expand $(\sqrt{u} + \sqrt{v}) (\sqrt{u} - \sqrt{v})$ using the FOIL Method.

Apply the distributive property.

$\sqrt{u} (\sqrt{u} - \sqrt{v}) + \sqrt{v} (\sqrt{u} - \sqrt{v})$

Apply the distributive property.

$\sqrt{u} \sqrt{u} + \sqrt{u} (- \sqrt{v}) + \sqrt{v} (\sqrt{u} - \sqrt{v})$

Apply the distributive property.

$\sqrt{u} \sqrt{u} + \sqrt{u} (- \sqrt{v}) + \sqrt{v} \sqrt{u} + \sqrt{v} (- \sqrt{v})$

Step 2

Simplify terms.

Combine the opposite terms in $\sqrt{u} \sqrt{u} + \sqrt{u} (- \sqrt{v}) + \sqrt{v} \sqrt{u} + \sqrt{v} (- \sqrt{v})$ .

Reorder the factors in the terms $\sqrt{u} (- \sqrt{v})$ and $\sqrt{v} \sqrt{u}$ .

$\sqrt{u} \sqrt{u} - \sqrt{u} \sqrt{v} + \sqrt{u} \sqrt{v} + \sqrt{v} (- \sqrt{v})$

Add $- \sqrt{u} \sqrt{v}$ and $\sqrt{u} \sqrt{v}$ .

$\sqrt{u} \sqrt{u} + 0 + \sqrt{v} (- \sqrt{v})$

Add $\sqrt{u} \sqrt{u}$ and $0$ .

$\sqrt{u} \sqrt{u} + \sqrt{v} (- \sqrt{v})$

Simplify each term.

Multiply $\sqrt{u} \sqrt{u}$ .

Raise $\sqrt{u}$ to the power of $1$ .

${\sqrt{u}}^{1} \sqrt{u} + \sqrt{v} (- \sqrt{v})$

Raise $\sqrt{u}$ to the power of $1$ .

${\sqrt{u}}^{1} {\sqrt{u}}^{1} + \sqrt{v} (- \sqrt{v})$

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

${\sqrt{u}}^{1 + 1} + \sqrt{v} (- \sqrt{v})$

Add $1$ and $1$ .

${\sqrt{u}}^{2} + \sqrt{v} (- \sqrt{v})$

Rewrite ${\sqrt{u}}^{2}$ as $u$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{u}$ as $u^{\frac{1}{2}}$ .

${(u^{\frac{1}{2}})}^{2} + \sqrt{v} (- \sqrt{v})$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$u^{\frac{1}{2} \cdot 2} + \sqrt{v} (- \sqrt{v})$

Combine $\frac{1}{2}$ and $2$ .

$u^{\frac{2}{2}} + \sqrt{v} (- \sqrt{v})$

Cancel the common factor of $2$ .

Cancel the common factor.

$u^{\frac{2}{2}} + \sqrt{v} (- \sqrt{v})$

Rewrite the expression.

$u^{1} + \sqrt{v} (- \sqrt{v})$

Simplify.

$u + \sqrt{v} (- \sqrt{v})$

Rewrite using the commutative property of multiplication.

$u - \sqrt{v} \sqrt{v}$

Multiply $- \sqrt{v} \sqrt{v}$ .

Raise $\sqrt{v}$ to the power of $1$ .

$u - ({\sqrt{v}}^{1} \sqrt{v})$

Raise $\sqrt{v}$ to the power of $1$ .

$u - ({\sqrt{v}}^{1} {\sqrt{v}}^{1})$

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

$u - {\sqrt{v}}^{1 + 1}$

Add $1$ and $1$ .

$u - {\sqrt{v}}^{2}$

Rewrite ${\sqrt{v}}^{2}$ as $v$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{v}$ as $v^{\frac{1}{2}}$ .

$u - {(v^{\frac{1}{2}})}^{2}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$u - v^{\frac{1}{2} \cdot 2}$

Combine $\frac{1}{2}$ and $2$ .

$u - v^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$u - v^{\frac{2}{2}}$

Rewrite the expression.

$u - v^{1}$

Simplify.

$u - v$

Simplify ( square root of u+ square root of v)( square root of u- square root of v)

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