Simplify ( square root of c- square root of d)/( square root of c+ square root of d)

$\frac{\sqrt{c} - \sqrt{d}}{\sqrt{c} + \sqrt{d}}$

Step 1

Multiply $\frac{\sqrt{c} - \sqrt{d}}{\sqrt{c} + \sqrt{d}}$ by $\frac{\sqrt{c} - \sqrt{d}}{\sqrt{c} - \sqrt{d}}$ .

$\frac{\sqrt{c} - \sqrt{d}}{\sqrt{c} + \sqrt{d}} \cdot \frac{\sqrt{c} - \sqrt{d}}{\sqrt{c} - \sqrt{d}}$

Step 2

Combine fractions.

Multiply $\frac{\sqrt{c} - \sqrt{d}}{\sqrt{c} + \sqrt{d}}$ by $\frac{\sqrt{c} - \sqrt{d}}{\sqrt{c} - \sqrt{d}}$ .

$\frac{(\sqrt{c} - \sqrt{d}) (\sqrt{c} - \sqrt{d})}{(\sqrt{c} + \sqrt{d}) (\sqrt{c} - \sqrt{d})}$

Expand the denominator using the FOIL method.

$\frac{(\sqrt{c} - \sqrt{d}) (\sqrt{c} - \sqrt{d})}{{\sqrt{c}}^{2} - \sqrt{c d} + \sqrt{d c} - {\sqrt{d}}^{2}}$

Simplify.

$\frac{(\sqrt{c} - \sqrt{d}) (\sqrt{c} - \sqrt{d})}{c - d}$

Step 3

Simplify the numerator.

Raise $\sqrt{c} - \sqrt{d}$ to the power of $1$ .

$\frac{{(\sqrt{c} - \sqrt{d})}^{1} (\sqrt{c} - \sqrt{d})}{c - d}$

Raise $\sqrt{c} - \sqrt{d}$ to the power of $1$ .

$\frac{{(\sqrt{c} - \sqrt{d})}^{1} {(\sqrt{c} - \sqrt{d})}^{1}}{c - d}$

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

$\frac{{(\sqrt{c} - \sqrt{d})}^{1 + 1}}{c - d}$

Add $1$ and $1$ .

$\frac{{(\sqrt{c} - \sqrt{d})}^{2}}{c - d}$

Step 4

Rewrite ${(\sqrt{c} - \sqrt{d})}^{2}$ as $(\sqrt{c} - \sqrt{d}) (\sqrt{c} - \sqrt{d})$ .

$\frac{(\sqrt{c} - \sqrt{d}) (\sqrt{c} - \sqrt{d})}{c - d}$

Step 5

Expand $(\sqrt{c} - \sqrt{d}) (\sqrt{c} - \sqrt{d})$ using the FOIL Method.

Apply the distributive property.

$\frac{\sqrt{c} (\sqrt{c} - \sqrt{d}) - \sqrt{d} (\sqrt{c} - \sqrt{d})}{c - d}$

Apply the distributive property.

$\frac{\sqrt{c} \sqrt{c} + \sqrt{c} (- \sqrt{d}) - \sqrt{d} (\sqrt{c} - \sqrt{d})}{c - d}$

Apply the distributive property.

$\frac{\sqrt{c} \sqrt{c} + \sqrt{c} (- \sqrt{d}) - \sqrt{d} \sqrt{c} - \sqrt{d} (- \sqrt{d})}{c - d}$

Step 6

Simplify and combine like terms.

Simplify each term.

Multiply $\sqrt{c} \sqrt{c}$ .

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

$\frac{{\sqrt{c}}^{1} \sqrt{c} + \sqrt{c} (- \sqrt{d}) - \sqrt{d} \sqrt{c} - \sqrt{d} (- \sqrt{d})}{c - d}$

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

$\frac{{\sqrt{c}}^{1} {\sqrt{c}}^{1} + \sqrt{c} (- \sqrt{d}) - \sqrt{d} \sqrt{c} - \sqrt{d} (- \sqrt{d})}{c - d}$

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

$\frac{{\sqrt{c}}^{1 + 1} + \sqrt{c} (- \sqrt{d}) - \sqrt{d} \sqrt{c} - \sqrt{d} (- \sqrt{d})}{c - d}$

Add $1$ and $1$ .

$\frac{{\sqrt{c}}^{2} + \sqrt{c} (- \sqrt{d}) - \sqrt{d} \sqrt{c} - \sqrt{d} (- \sqrt{d})}{c - d}$

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

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

$\frac{{(c^{\frac{1}{2}})}^{2} + \sqrt{c} (- \sqrt{d}) - \sqrt{d} \sqrt{c} - \sqrt{d} (- \sqrt{d})}{c - d}$

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

$\frac{c^{\frac{1}{2} \cdot 2} + \sqrt{c} (- \sqrt{d}) - \sqrt{d} \sqrt{c} - \sqrt{d} (- \sqrt{d})}{c - d}$

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

$\frac{c^{\frac{2}{2}} + \sqrt{c} (- \sqrt{d}) - \sqrt{d} \sqrt{c} - \sqrt{d} (- \sqrt{d})}{c - d}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{c^{\frac{2}{2}} + \sqrt{c} (- \sqrt{d}) - \sqrt{d} \sqrt{c} - \sqrt{d} (- \sqrt{d})}{c - d}$

Rewrite the expression.

$\frac{c^{1} + \sqrt{c} (- \sqrt{d}) - \sqrt{d} \sqrt{c} - \sqrt{d} (- \sqrt{d})}{c - d}$

Simplify.

$\frac{c + \sqrt{c} (- \sqrt{d}) - \sqrt{d} \sqrt{c} - \sqrt{d} (- \sqrt{d})}{c - d}$

Rewrite using the commutative property of multiplication.

$\frac{c - \sqrt{c} \sqrt{d} - \sqrt{d} \sqrt{c} - \sqrt{d} (- \sqrt{d})}{c - d}$

Combine using the product rule for radicals.

$\frac{c - \sqrt{d c} - \sqrt{d} \sqrt{c} - \sqrt{d} (- \sqrt{d})}{c - d}$

Combine using the product rule for radicals.

$\frac{c - \sqrt{d c} - \sqrt{c d} - \sqrt{d} (- \sqrt{d})}{c - d}$

Multiply $- \sqrt{d} (- \sqrt{d})$ .

Multiply $- 1$ by $- 1$ .

$\frac{c - \sqrt{d c} - \sqrt{c d} + 1 \sqrt{d} \sqrt{d}}{c - d}$

Multiply $\sqrt{d}$ by $1$ .

$\frac{c - \sqrt{d c} - \sqrt{c d} + \sqrt{d} \sqrt{d}}{c - d}$

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

$\frac{c - \sqrt{d c} - \sqrt{c d} + {\sqrt{d}}^{1} \sqrt{d}}{c - d}$

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

$\frac{c - \sqrt{d c} - \sqrt{c d} + {\sqrt{d}}^{1} {\sqrt{d}}^{1}}{c - d}$

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

$\frac{c - \sqrt{d c} - \sqrt{c d} + {\sqrt{d}}^{1 + 1}}{c - d}$

Add $1$ and $1$ .

$\frac{c - \sqrt{d c} - \sqrt{c d} + {\sqrt{d}}^{2}}{c - d}$

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

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

$\frac{c - \sqrt{d c} - \sqrt{c d} + {(d^{\frac{1}{2}})}^{2}}{c - d}$

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

$\frac{c - \sqrt{d c} - \sqrt{c d} + d^{\frac{1}{2} \cdot 2}}{c - d}$

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

$\frac{c - \sqrt{d c} - \sqrt{c d} + d^{\frac{2}{2}}}{c - d}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{c - \sqrt{d c} - \sqrt{c d} + d^{\frac{2}{2}}}{c - d}$

Rewrite the expression.

$\frac{c - \sqrt{d c} - \sqrt{c d} + d^{1}}{c - d}$

Simplify.

$\frac{c - \sqrt{d c} - \sqrt{c d} + d}{c - d}$

Subtract $\sqrt{c d}$ from $- \sqrt{d c}$ .

Reorder $d$ and $c$ .

$\frac{c - \sqrt{c d} - \sqrt{c d} + d}{c - d}$

Subtract $\sqrt{c d}$ from $- \sqrt{c d}$ .

$\frac{c - 2 \sqrt{c d} + d}{c - d}$

Simplify ( square root of c- square root of d)/( square root of c+ square root of d)

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