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

$\frac{\sqrt{u}}{\sqrt{u} + \sqrt{v}}$

Step 1

Multiply $\frac{\sqrt{u}}{\sqrt{u} + \sqrt{v}}$ by $\frac{\sqrt{u} - \sqrt{v}}{\sqrt{u} - \sqrt{v}}$ .

$\frac{\sqrt{u}}{\sqrt{u} + \sqrt{v}} \cdot \frac{\sqrt{u} - \sqrt{v}}{\sqrt{u} - \sqrt{v}}$

Step 2

Multiply $\frac{\sqrt{u}}{\sqrt{u} + \sqrt{v}}$ by $\frac{\sqrt{u} - \sqrt{v}}{\sqrt{u} - \sqrt{v}}$ .

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

Step 3

Expand the denominator using the FOIL method.

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

Step 4

Simplify.

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

Step 5

Apply the distributive property.

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

Step 6

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

Step 7

Simplify terms.

Rewrite using the commutative property of multiplication.

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

Simplify each term.

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Combine using the product rule for radicals.

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

Step 8

Simplify the numerator.

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

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

Apply the product rule to $v u$ .

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

Factor $u^{\frac{1}{2}}$ out of $u - v^{\frac{1}{2}} u^{\frac{1}{2}}$ .

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

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

Raise $u$ to the power of $1$ .

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

Factor $u^{\frac{1}{2}}$ out of $u^{1}$ .

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

Factor $u^{\frac{1}{2}}$ out of $- u^{\frac{1}{2}} v^{\frac{1}{2}}$ .

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

Factor $u^{\frac{1}{2}}$ out of $u^{\frac{1}{2}} u^{\frac{1}{2}} + u^{\frac{1}{2}} (- v^{\frac{1}{2}})$ .

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

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

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