Simplify ( square root of r- square root of s)/( square root of r+ square root of s)

$\frac{\sqrt{r} - \sqrt{s}}{\sqrt{r} + \sqrt{s}}$

Step 1

Multiply $\frac{\sqrt{r} - \sqrt{s}}{\sqrt{r} + \sqrt{s}}$ by $\frac{\sqrt{r} - \sqrt{s}}{\sqrt{r} - \sqrt{s}}$ .

$\frac{\sqrt{r} - \sqrt{s}}{\sqrt{r} + \sqrt{s}} \cdot \frac{\sqrt{r} - \sqrt{s}}{\sqrt{r} - \sqrt{s}}$

Step 2

Combine fractions.

Multiply $\frac{\sqrt{r} - \sqrt{s}}{\sqrt{r} + \sqrt{s}}$ by $\frac{\sqrt{r} - \sqrt{s}}{\sqrt{r} - \sqrt{s}}$ .

$\frac{(\sqrt{r} - \sqrt{s}) (\sqrt{r} - \sqrt{s})}{(\sqrt{r} + \sqrt{s}) (\sqrt{r} - \sqrt{s})}$

Expand the denominator using the FOIL method.

$\frac{(\sqrt{r} - \sqrt{s}) (\sqrt{r} - \sqrt{s})}{{\sqrt{r}}^{2} - \sqrt{r s} + \sqrt{s r} - {\sqrt{s}}^{2}}$

Simplify.

$\frac{(\sqrt{r} - \sqrt{s}) (\sqrt{r} - \sqrt{s})}{r - s}$

Step 3

Simplify the numerator.

Raise $\sqrt{r} - \sqrt{s}$ to the power of $1$ .

$\frac{{(\sqrt{r} - \sqrt{s})}^{1} (\sqrt{r} - \sqrt{s})}{r - s}$

Raise $\sqrt{r} - \sqrt{s}$ to the power of $1$ .

$\frac{{(\sqrt{r} - \sqrt{s})}^{1} {(\sqrt{r} - \sqrt{s})}^{1}}{r - s}$

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

$\frac{{(\sqrt{r} - \sqrt{s})}^{1 + 1}}{r - s}$

Add $1$ and $1$ .

$\frac{{(\sqrt{r} - \sqrt{s})}^{2}}{r - s}$

Step 4

Rewrite ${(\sqrt{r} - \sqrt{s})}^{2}$ as $(\sqrt{r} - \sqrt{s}) (\sqrt{r} - \sqrt{s})$ .

$\frac{(\sqrt{r} - \sqrt{s}) (\sqrt{r} - \sqrt{s})}{r - s}$

Step 5

Expand $(\sqrt{r} - \sqrt{s}) (\sqrt{r} - \sqrt{s})$ using the FOIL Method.

Apply the distributive property.

$\frac{\sqrt{r} (\sqrt{r} - \sqrt{s}) - \sqrt{s} (\sqrt{r} - \sqrt{s})}{r - s}$

Apply the distributive property.

$\frac{\sqrt{r} \sqrt{r} + \sqrt{r} (- \sqrt{s}) - \sqrt{s} (\sqrt{r} - \sqrt{s})}{r - s}$

Apply the distributive property.

$\frac{\sqrt{r} \sqrt{r} + \sqrt{r} (- \sqrt{s}) - \sqrt{s} \sqrt{r} - \sqrt{s} (- \sqrt{s})}{r - s}$

Step 6

Simplify and combine like terms.

Simplify each term.

Multiply $\sqrt{r} \sqrt{r}$ .

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

$\frac{{\sqrt{r}}^{1} \sqrt{r} + \sqrt{r} (- \sqrt{s}) - \sqrt{s} \sqrt{r} - \sqrt{s} (- \sqrt{s})}{r - s}$

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

$\frac{{\sqrt{r}}^{1} {\sqrt{r}}^{1} + \sqrt{r} (- \sqrt{s}) - \sqrt{s} \sqrt{r} - \sqrt{s} (- \sqrt{s})}{r - s}$

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

$\frac{{\sqrt{r}}^{1 + 1} + \sqrt{r} (- \sqrt{s}) - \sqrt{s} \sqrt{r} - \sqrt{s} (- \sqrt{s})}{r - s}$

Add $1$ and $1$ .

$\frac{{\sqrt{r}}^{2} + \sqrt{r} (- \sqrt{s}) - \sqrt{s} \sqrt{r} - \sqrt{s} (- \sqrt{s})}{r - s}$

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

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

$\frac{{(r^{\frac{1}{2}})}^{2} + \sqrt{r} (- \sqrt{s}) - \sqrt{s} \sqrt{r} - \sqrt{s} (- \sqrt{s})}{r - s}$

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

$\frac{r^{\frac{1}{2} \cdot 2} + \sqrt{r} (- \sqrt{s}) - \sqrt{s} \sqrt{r} - \sqrt{s} (- \sqrt{s})}{r - s}$

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

$\frac{r^{\frac{2}{2}} + \sqrt{r} (- \sqrt{s}) - \sqrt{s} \sqrt{r} - \sqrt{s} (- \sqrt{s})}{r - s}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{r^{\frac{2}{2}} + \sqrt{r} (- \sqrt{s}) - \sqrt{s} \sqrt{r} - \sqrt{s} (- \sqrt{s})}{r - s}$

Rewrite the expression.

$\frac{r^{1} + \sqrt{r} (- \sqrt{s}) - \sqrt{s} \sqrt{r} - \sqrt{s} (- \sqrt{s})}{r - s}$

Simplify.

$\frac{r + \sqrt{r} (- \sqrt{s}) - \sqrt{s} \sqrt{r} - \sqrt{s} (- \sqrt{s})}{r - s}$

Rewrite using the commutative property of multiplication.

$\frac{r - \sqrt{r} \sqrt{s} - \sqrt{s} \sqrt{r} - \sqrt{s} (- \sqrt{s})}{r - s}$

Combine using the product rule for radicals.

$\frac{r - \sqrt{s r} - \sqrt{s} \sqrt{r} - \sqrt{s} (- \sqrt{s})}{r - s}$

Combine using the product rule for radicals.

$\frac{r - \sqrt{s r} - \sqrt{r s} - \sqrt{s} (- \sqrt{s})}{r - s}$

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

Multiply $- 1$ by $- 1$ .

$\frac{r - \sqrt{s r} - \sqrt{r s} + 1 \sqrt{s} \sqrt{s}}{r - s}$

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

$\frac{r - \sqrt{s r} - \sqrt{r s} + \sqrt{s} \sqrt{s}}{r - s}$

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

$\frac{r - \sqrt{s r} - \sqrt{r s} + {\sqrt{s}}^{1} \sqrt{s}}{r - s}$

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

$\frac{r - \sqrt{s r} - \sqrt{r s} + {\sqrt{s}}^{1} {\sqrt{s}}^{1}}{r - s}$

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

$\frac{r - \sqrt{s r} - \sqrt{r s} + {\sqrt{s}}^{1 + 1}}{r - s}$

Add $1$ and $1$ .

$\frac{r - \sqrt{s r} - \sqrt{r s} + {\sqrt{s}}^{2}}{r - s}$

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

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

$\frac{r - \sqrt{s r} - \sqrt{r s} + {(s^{\frac{1}{2}})}^{2}}{r - s}$

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

$\frac{r - \sqrt{s r} - \sqrt{r s} + s^{\frac{1}{2} \cdot 2}}{r - s}$

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

$\frac{r - \sqrt{s r} - \sqrt{r s} + s^{\frac{2}{2}}}{r - s}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{r - \sqrt{s r} - \sqrt{r s} + s^{\frac{2}{2}}}{r - s}$

Rewrite the expression.

$\frac{r - \sqrt{s r} - \sqrt{r s} + s^{1}}{r - s}$

Simplify.

$\frac{r - \sqrt{s r} - \sqrt{r s} + s}{r - s}$

Subtract $\sqrt{r s}$ from $- \sqrt{s r}$ .

Reorder $s$ and $r$ .

$\frac{r - \sqrt{r s} - \sqrt{r s} + s}{r - s}$

Subtract $\sqrt{r s}$ from $- \sqrt{r s}$ .

$\frac{r - 2 \sqrt{r s} + s}{r - s}$

Simplify ( square root of r- square root of s)/( square root of r+ square root of s)

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