Finite Math Examples

$\frac{\sqrt{2 a} - \sqrt{b}}{\sqrt{2 a} + \sqrt{b}} - \frac{\sqrt{2 a} + \sqrt{b}}{\sqrt{2 a} - \sqrt{b}}$

Step 1

Simplify each term.

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Step 1.1

Multiply $\frac{\sqrt{2 a} - \sqrt{b}}{\sqrt{2 a} + \sqrt{b}}$ by $\frac{\sqrt{2 a} - \sqrt{b}}{\sqrt{2 a} - \sqrt{b}}$ .

$\frac{\sqrt{2 a} - \sqrt{b}}{\sqrt{2 a} + \sqrt{b}} \cdot \frac{\sqrt{2 a} - \sqrt{b}}{\sqrt{2 a} - \sqrt{b}} - \frac{\sqrt{2 a} + \sqrt{b}}{\sqrt{2 a} - \sqrt{b}}$

Step 1.2

Multiply $\frac{\sqrt{2 a} - \sqrt{b}}{\sqrt{2 a} + \sqrt{b}}$ by $\frac{\sqrt{2 a} - \sqrt{b}}{\sqrt{2 a} - \sqrt{b}}$ .

$\frac{(\sqrt{2 a} - \sqrt{b}) (\sqrt{2 a} - \sqrt{b})}{(\sqrt{2 a} + \sqrt{b}) (\sqrt{2 a} - \sqrt{b})} - \frac{\sqrt{2 a} + \sqrt{b}}{\sqrt{2 a} - \sqrt{b}}$

Step 1.3

Expand the denominator using the FOIL method.

$\frac{(\sqrt{2 a} - \sqrt{b}) (\sqrt{2 a} - \sqrt{b})}{{\sqrt{2 a}}^{2} - \sqrt{2 a b} + \sqrt{b \cdot 2 a} - {\sqrt{b}}^{2}} - \frac{\sqrt{2 a} + \sqrt{b}}{\sqrt{2 a} - \sqrt{b}}$

Step 1.4

Simplify.

$\frac{(\sqrt{2 a} - \sqrt{b}) (\sqrt{2 a} - \sqrt{b})}{2 a - b} - \frac{\sqrt{2 a} + \sqrt{b}}{\sqrt{2 a} - \sqrt{b}}$

Step 1.5

Simplify the numerator.

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Step 1.5.1

Raise $\sqrt{2 a} - \sqrt{b}$ to the power of $1$ .

$\frac{{(\sqrt{2 a} - \sqrt{b})}^{1} (\sqrt{2 a} - \sqrt{b})}{2 a - b} - \frac{\sqrt{2 a} + \sqrt{b}}{\sqrt{2 a} - \sqrt{b}}$

Step 1.5.2

Raise $\sqrt{2 a} - \sqrt{b}$ to the power of $1$ .

$\frac{{(\sqrt{2 a} - \sqrt{b})}^{1} {(\sqrt{2 a} - \sqrt{b})}^{1}}{2 a - b} - \frac{\sqrt{2 a} + \sqrt{b}}{\sqrt{2 a} - \sqrt{b}}$

Step 1.5.3

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

$\frac{{(\sqrt{2 a} - \sqrt{b})}^{1 + 1}}{2 a - b} - \frac{\sqrt{2 a} + \sqrt{b}}{\sqrt{2 a} - \sqrt{b}}$

Step 1.5.4

Add $1$ and $1$ .

$\frac{{(\sqrt{2 a} - \sqrt{b})}^{2}}{2 a - b} - \frac{\sqrt{2 a} + \sqrt{b}}{\sqrt{2 a} - \sqrt{b}}$

Step 1.6

Rewrite ${(\sqrt{2 a} - \sqrt{b})}^{2}$ as $(\sqrt{2 a} - \sqrt{b}) (\sqrt{2 a} - \sqrt{b})$ .

$\frac{(\sqrt{2 a} - \sqrt{b}) (\sqrt{2 a} - \sqrt{b})}{2 a - b} - \frac{\sqrt{2 a} + \sqrt{b}}{\sqrt{2 a} - \sqrt{b}}$

Step 1.7

Expand $(\sqrt{2 a} - \sqrt{b}) (\sqrt{2 a} - \sqrt{b})$ using the FOIL Method.

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Step 1.7.1

Apply the distributive property.

$\frac{\sqrt{2 a} (\sqrt{2 a} - \sqrt{b}) - \sqrt{b} (\sqrt{2 a} - \sqrt{b})}{2 a - b} - \frac{\sqrt{2 a} + \sqrt{b}}{\sqrt{2 a} - \sqrt{b}}$

Step 1.7.2

Apply the distributive property.

$\frac{\sqrt{2 a} \sqrt{2 a} + \sqrt{2 a} (- \sqrt{b}) - \sqrt{b} (\sqrt{2 a} - \sqrt{b})}{2 a - b} - \frac{\sqrt{2 a} + \sqrt{b}}{\sqrt{2 a} - \sqrt{b}}$

Step 1.7.3

Apply the distributive property.

$\frac{\sqrt{2 a} \sqrt{2 a} + \sqrt{2 a} (- \sqrt{b}) - \sqrt{b} \sqrt{2 a} - \sqrt{b} (- \sqrt{b})}{2 a - b} - \frac{\sqrt{2 a} + \sqrt{b}}{\sqrt{2 a} - \sqrt{b}}$

Step 1.8

Simplify and combine like terms.

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Step 1.8.1

Simplify each term.

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Step 1.8.1.1

Multiply $\sqrt{2 a} \sqrt{2 a}$ .

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Step 1.8.1.1.1

Raise $\sqrt{2 a}$ to the power of $1$ .

$\frac{{\sqrt{2 a}}^{1} \sqrt{2 a} + \sqrt{2 a} (- \sqrt{b}) - \sqrt{b} \sqrt{2 a} - \sqrt{b} (- \sqrt{b})}{2 a - b} - \frac{\sqrt{2 a} + \sqrt{b}}{\sqrt{2 a} - \sqrt{b}}$

Step 1.8.1.1.2

Raise $\sqrt{2 a}$ to the power of $1$ .

$\frac{{\sqrt{2 a}}^{1} {\sqrt{2 a}}^{1} + \sqrt{2 a} (- \sqrt{b}) - \sqrt{b} \sqrt{2 a} - \sqrt{b} (- \sqrt{b})}{2 a - b} - \frac{\sqrt{2 a} + \sqrt{b}}{\sqrt{2 a} - \sqrt{b}}$

Step 1.8.1.1.3

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

$\frac{{\sqrt{2 a}}^{1 + 1} + \sqrt{2 a} (- \sqrt{b}) - \sqrt{b} \sqrt{2 a} - \sqrt{b} (- \sqrt{b})}{2 a - b} - \frac{\sqrt{2 a} + \sqrt{b}}{\sqrt{2 a} - \sqrt{b}}$

Step 1.8.1.1.4

Add $1$ and $1$ .

$\frac{{\sqrt{2 a}}^{2} + \sqrt{2 a} (- \sqrt{b}) - \sqrt{b} \sqrt{2 a} - \sqrt{b} (- \sqrt{b})}{2 a - b} - \frac{\sqrt{2 a} + \sqrt{b}}{\sqrt{2 a} - \sqrt{b}}$

Step 1.8.1.2

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

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Step 1.8.1.2.1

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

$\frac{{({(2 a)}^{\frac{1}{2}})}^{2} + \sqrt{2 a} (- \sqrt{b}) - \sqrt{b} \sqrt{2 a} - \sqrt{b} (- \sqrt{b})}{2 a - b} - \frac{\sqrt{2 a} + \sqrt{b}}{\sqrt{2 a} - \sqrt{b}}$

Step 1.8.1.2.2

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

$\frac{{(2 a)}^{\frac{1}{2} \cdot 2} + \sqrt{2 a} (- \sqrt{b}) - \sqrt{b} \sqrt{2 a} - \sqrt{b} (- \sqrt{b})}{2 a - b} - \frac{\sqrt{2 a} + \sqrt{b}}{\sqrt{2 a} - \sqrt{b}}$

Step 1.8.1.2.3

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

$\frac{{(2 a)}^{\frac{2}{2}} + \sqrt{2 a} (- \sqrt{b}) - \sqrt{b} \sqrt{2 a} - \sqrt{b} (- \sqrt{b})}{2 a - b} - \frac{\sqrt{2 a} + \sqrt{b}}{\sqrt{2 a} - \sqrt{b}}$

Step 1.8.1.2.4

Cancel the common factor of $2$ .

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Step 1.8.1.2.4.1

Cancel the common factor.

$\frac{{(2 a)}^{\frac{2}{2}} + \sqrt{2 a} (- \sqrt{b}) - \sqrt{b} \sqrt{2 a} - \sqrt{b} (- \sqrt{b})}{2 a - b} - \frac{\sqrt{2 a} + \sqrt{b}}{\sqrt{2 a} - \sqrt{b}}$

Step 1.8.1.2.4.2

Rewrite the expression.

$\frac{{(2 a)}^{1} + \sqrt{2 a} (- \sqrt{b}) - \sqrt{b} \sqrt{2 a} - \sqrt{b} (- \sqrt{b})}{2 a - b} - \frac{\sqrt{2 a} + \sqrt{b}}{\sqrt{2 a} - \sqrt{b}}$

Step 1.8.1.2.5

Simplify.

$\frac{2 a + \sqrt{2 a} (- \sqrt{b}) - \sqrt{b} \sqrt{2 a} - \sqrt{b} (- \sqrt{b})}{2 a - b} - \frac{\sqrt{2 a} + \sqrt{b}}{\sqrt{2 a} - \sqrt{b}}$

Step 1.8.1.3

Rewrite using the commutative property of multiplication.

$\frac{2 a - \sqrt{2 a} \sqrt{b} - \sqrt{b} \sqrt{2 a} - \sqrt{b} (- \sqrt{b})}{2 a - b} - \frac{\sqrt{2 a} + \sqrt{b}}{\sqrt{2 a} - \sqrt{b}}$

Step 1.8.1.4

Combine using the product rule for radicals.

$\frac{2 a - \sqrt{b \cdot 2 a} - \sqrt{b} \sqrt{2 a} - \sqrt{b} (- \sqrt{b})}{2 a - b} - \frac{\sqrt{2 a} + \sqrt{b}}{\sqrt{2 a} - \sqrt{b}}$

Step 1.8.1.5

Combine using the product rule for radicals.

$\frac{2 a - \sqrt{b \cdot 2 a} - \sqrt{2 a b} - \sqrt{b} (- \sqrt{b})}{2 a - b} - \frac{\sqrt{2 a} + \sqrt{b}}{\sqrt{2 a} - \sqrt{b}}$

Step 1.8.1.6

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

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Step 1.8.1.6.1

Multiply $- 1$ by $- 1$ .

$\frac{2 a - \sqrt{b \cdot 2 a} - \sqrt{2 a b} + 1 \sqrt{b} \sqrt{b}}{2 a - b} - \frac{\sqrt{2 a} + \sqrt{b}}{\sqrt{2 a} - \sqrt{b}}$

Step 1.8.1.6.2

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

$\frac{2 a - \sqrt{b \cdot 2 a} - \sqrt{2 a b} + \sqrt{b} \sqrt{b}}{2 a - b} - \frac{\sqrt{2 a} + \sqrt{b}}{\sqrt{2 a} - \sqrt{b}}$

Step 1.8.1.6.3

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

$\frac{2 a - \sqrt{b \cdot 2 a} - \sqrt{2 a b} + {\sqrt{b}}^{1} \sqrt{b}}{2 a - b} - \frac{\sqrt{2 a} + \sqrt{b}}{\sqrt{2 a} - \sqrt{b}}$

Step 1.8.1.6.4

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

$\frac{2 a - \sqrt{b \cdot 2 a} - \sqrt{2 a b} + {\sqrt{b}}^{1} {\sqrt{b}}^{1}}{2 a - b} - \frac{\sqrt{2 a} + \sqrt{b}}{\sqrt{2 a} - \sqrt{b}}$

Step 1.8.1.6.5

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

$\frac{2 a - \sqrt{b \cdot 2 a} - \sqrt{2 a b} + {\sqrt{b}}^{1 + 1}}{2 a - b} - \frac{\sqrt{2 a} + \sqrt{b}}{\sqrt{2 a} - \sqrt{b}}$

Step 1.8.1.6.6

Add $1$ and $1$ .

$\frac{2 a - \sqrt{b \cdot 2 a} - \sqrt{2 a b} + {\sqrt{b}}^{2}}{2 a - b} - \frac{\sqrt{2 a} + \sqrt{b}}{\sqrt{2 a} - \sqrt{b}}$

Step 1.8.1.7

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

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Step 1.8.1.7.1

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

$\frac{2 a - \sqrt{b \cdot 2 a} - \sqrt{2 a b} + {(b^{\frac{1}{2}})}^{2}}{2 a - b} - \frac{\sqrt{2 a} + \sqrt{b}}{\sqrt{2 a} - \sqrt{b}}$

Step 1.8.1.7.2

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

$\frac{2 a - \sqrt{b \cdot 2 a} - \sqrt{2 a b} + b^{\frac{1}{2} \cdot 2}}{2 a - b} - \frac{\sqrt{2 a} + \sqrt{b}}{\sqrt{2 a} - \sqrt{b}}$

Step 1.8.1.7.3

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

$\frac{2 a - \sqrt{b \cdot 2 a} - \sqrt{2 a b} + b^{\frac{2}{2}}}{2 a - b} - \frac{\sqrt{2 a} + \sqrt{b}}{\sqrt{2 a} - \sqrt{b}}$

Step 1.8.1.7.4

Cancel the common factor of $2$ .

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Step 1.8.1.7.4.1

Cancel the common factor.

$\frac{2 a - \sqrt{b \cdot 2 a} - \sqrt{2 a b} + b^{\frac{2}{2}}}{2 a - b} - \frac{\sqrt{2 a} + \sqrt{b}}{\sqrt{2 a} - \sqrt{b}}$

Step 1.8.1.7.4.2

Rewrite the expression.

$\frac{2 a - \sqrt{b \cdot 2 a} - \sqrt{2 a b} + b^{1}}{2 a - b} - \frac{\sqrt{2 a} + \sqrt{b}}{\sqrt{2 a} - \sqrt{b}}$

Step 1.8.1.7.5

Simplify.

$\frac{2 a - \sqrt{b \cdot 2 a} - \sqrt{2 a b} + b}{2 a - b} - \frac{\sqrt{2 a} + \sqrt{b}}{\sqrt{2 a} - \sqrt{b}}$

Step 1.8.2

Subtract $\sqrt{2 a b}$ from $- \sqrt{b \cdot 2 a}$ .

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Step 1.8.2.1

Move $b$ .

$\frac{2 a - \sqrt{2 a b} - \sqrt{2 a b} + b}{2 a - b} - \frac{\sqrt{2 a} + \sqrt{b}}{\sqrt{2 a} - \sqrt{b}}$

Step 1.8.2.2

Subtract $\sqrt{2 a b}$ from $- \sqrt{2 a b}$ .

$\frac{2 a - 2 \sqrt{2 a b} + b}{2 a - b} - \frac{\sqrt{2 a} + \sqrt{b}}{\sqrt{2 a} - \sqrt{b}}$

Step 1.9

Multiply $\frac{\sqrt{2 a} + \sqrt{b}}{\sqrt{2 a} - \sqrt{b}}$ by $\frac{\sqrt{2 a} + \sqrt{b}}{\sqrt{2 a} + \sqrt{b}}$ .

$\frac{2 a - 2 \sqrt{2 a b} + b}{2 a - b} - (\frac{\sqrt{2 a} + \sqrt{b}}{\sqrt{2 a} - \sqrt{b}} \cdot \frac{\sqrt{2 a} + \sqrt{b}}{\sqrt{2 a} + \sqrt{b}})$

Step 1.10

Multiply $\frac{\sqrt{2 a} + \sqrt{b}}{\sqrt{2 a} - \sqrt{b}}$ by $\frac{\sqrt{2 a} + \sqrt{b}}{\sqrt{2 a} + \sqrt{b}}$ .

$\frac{2 a - 2 \sqrt{2 a b} + b}{2 a - b} - \frac{(\sqrt{2 a} + \sqrt{b}) (\sqrt{2 a} + \sqrt{b})}{(\sqrt{2 a} - \sqrt{b}) (\sqrt{2 a} + \sqrt{b})}$

Step 1.11

Expand the denominator using the FOIL method.

$\frac{2 a - 2 \sqrt{2 a b} + b}{2 a - b} - \frac{(\sqrt{2 a} + \sqrt{b}) (\sqrt{2 a} + \sqrt{b})}{{\sqrt{2 a}}^{2} + \sqrt{2 a b} - \sqrt{2 a b} - {\sqrt{b}}^{2}}$

Step 1.12

Simplify.

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Step 1.12.1

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

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Step 1.12.1.1

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

$\frac{2 a - 2 \sqrt{2 a b} + b}{2 a - b} - \frac{(\sqrt{2 a} + \sqrt{b}) (\sqrt{2 a} + \sqrt{b})}{{({(2 a)}^{\frac{1}{2}})}^{2} - {\sqrt{b}}^{2}}$

Step 1.12.1.2

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

$\frac{2 a - 2 \sqrt{2 a b} + b}{2 a - b} - \frac{(\sqrt{2 a} + \sqrt{b}) (\sqrt{2 a} + \sqrt{b})}{{(2 a)}^{\frac{1}{2} \cdot 2} - {\sqrt{b}}^{2}}$

Step 1.12.1.3

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

$\frac{2 a - 2 \sqrt{2 a b} + b}{2 a - b} - \frac{(\sqrt{2 a} + \sqrt{b}) (\sqrt{2 a} + \sqrt{b})}{{(2 a)}^{\frac{2}{2}} - {\sqrt{b}}^{2}}$

Step 1.12.1.4

Cancel the common factor of $2$ .

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Step 1.12.1.4.1

Cancel the common factor.

$\frac{2 a - 2 \sqrt{2 a b} + b}{2 a - b} - \frac{(\sqrt{2 a} + \sqrt{b}) (\sqrt{2 a} + \sqrt{b})}{{(2 a)}^{\frac{2}{2}} - {\sqrt{b}}^{2}}$

Step 1.12.1.4.2

Rewrite the expression.

$\frac{2 a - 2 \sqrt{2 a b} + b}{2 a - b} - \frac{(\sqrt{2 a} + \sqrt{b}) (\sqrt{2 a} + \sqrt{b})}{{(2 a)}^{1} - {\sqrt{b}}^{2}}$

Step 1.12.1.5

Simplify.

$\frac{2 a - 2 \sqrt{2 a b} + b}{2 a - b} - \frac{(\sqrt{2 a} + \sqrt{b}) (\sqrt{2 a} + \sqrt{b})}{2 a - {\sqrt{b}}^{2}}$

Step 1.12.2

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

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Step 1.12.2.1

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

$\frac{2 a - 2 \sqrt{2 a b} + b}{2 a - b} - \frac{(\sqrt{2 a} + \sqrt{b}) (\sqrt{2 a} + \sqrt{b})}{2 a - {(b^{\frac{1}{2}})}^{2}}$

Step 1.12.2.2

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

$\frac{2 a - 2 \sqrt{2 a b} + b}{2 a - b} - \frac{(\sqrt{2 a} + \sqrt{b}) (\sqrt{2 a} + \sqrt{b})}{2 a - b^{\frac{1}{2} \cdot 2}}$

Step 1.12.2.3

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

$\frac{2 a - 2 \sqrt{2 a b} + b}{2 a - b} - \frac{(\sqrt{2 a} + \sqrt{b}) (\sqrt{2 a} + \sqrt{b})}{2 a - b^{\frac{2}{2}}}$

Step 1.12.2.4

Cancel the common factor of $2$ .

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Step 1.12.2.4.1

Cancel the common factor.

$\frac{2 a - 2 \sqrt{2 a b} + b}{2 a - b} - \frac{(\sqrt{2 a} + \sqrt{b}) (\sqrt{2 a} + \sqrt{b})}{2 a - b^{\frac{2}{2}}}$

Step 1.12.2.4.2

Rewrite the expression.

$\frac{2 a - 2 \sqrt{2 a b} + b}{2 a - b} - \frac{(\sqrt{2 a} + \sqrt{b}) (\sqrt{2 a} + \sqrt{b})}{2 a - b^{1}}$

Step 1.12.2.5

Simplify.

$\frac{2 a - 2 \sqrt{2 a b} + b}{2 a - b} - \frac{(\sqrt{2 a} + \sqrt{b}) (\sqrt{2 a} + \sqrt{b})}{2 a - b}$

Step 1.13

Simplify the numerator.

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Step 1.13.1

Raise $\sqrt{2 a} + \sqrt{b}$ to the power of $1$ .

$\frac{2 a - 2 \sqrt{2 a b} + b}{2 a - b} - \frac{{(\sqrt{2 a} + \sqrt{b})}^{1} (\sqrt{2 a} + \sqrt{b})}{2 a - b}$

Step 1.13.2

Raise $\sqrt{2 a} + \sqrt{b}$ to the power of $1$ .

$\frac{2 a - 2 \sqrt{2 a b} + b}{2 a - b} - \frac{{(\sqrt{2 a} + \sqrt{b})}^{1} {(\sqrt{2 a} + \sqrt{b})}^{1}}{2 a - b}$

Step 1.13.3

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

$\frac{2 a - 2 \sqrt{2 a b} + b}{2 a - b} - \frac{{(\sqrt{2 a} + \sqrt{b})}^{1 + 1}}{2 a - b}$

Step 1.13.4

Add $1$ and $1$ .

$\frac{2 a - 2 \sqrt{2 a b} + b}{2 a - b} - \frac{{(\sqrt{2 a} + \sqrt{b})}^{2}}{2 a - b}$

Step 1.14

Rewrite ${(\sqrt{2 a} + \sqrt{b})}^{2}$ as $(\sqrt{2 a} + \sqrt{b}) (\sqrt{2 a} + \sqrt{b})$ .

$\frac{2 a - 2 \sqrt{2 a b} + b}{2 a - b} - \frac{(\sqrt{2 a} + \sqrt{b}) (\sqrt{2 a} + \sqrt{b})}{2 a - b}$

Step 1.15

Expand $(\sqrt{2 a} + \sqrt{b}) (\sqrt{2 a} + \sqrt{b})$ using the FOIL Method.

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Step 1.15.1

Apply the distributive property.

$\frac{2 a - 2 \sqrt{2 a b} + b}{2 a - b} - \frac{\sqrt{2 a} (\sqrt{2 a} + \sqrt{b}) + \sqrt{b} (\sqrt{2 a} + \sqrt{b})}{2 a - b}$

Step 1.15.2

Apply the distributive property.

$\frac{2 a - 2 \sqrt{2 a b} + b}{2 a - b} - \frac{\sqrt{2 a} \sqrt{2 a} + \sqrt{2 a} \sqrt{b} + \sqrt{b} (\sqrt{2 a} + \sqrt{b})}{2 a - b}$

Step 1.15.3

Apply the distributive property.

$\frac{2 a - 2 \sqrt{2 a b} + b}{2 a - b} - \frac{\sqrt{2 a} \sqrt{2 a} + \sqrt{2 a} \sqrt{b} + \sqrt{b} \sqrt{2 a} + \sqrt{b} \sqrt{b}}{2 a - b}$

Step 1.16

Simplify and combine like terms.

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Step 1.16.1

Simplify each term.

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Step 1.16.1.1

Multiply $\sqrt{2 a} \sqrt{2 a}$ .

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Step 1.16.1.1.1

Raise $\sqrt{2 a}$ to the power of $1$ .

$\frac{2 a - 2 \sqrt{2 a b} + b}{2 a - b} - \frac{{\sqrt{2 a}}^{1} \sqrt{2 a} + \sqrt{2 a} \sqrt{b} + \sqrt{b} \sqrt{2 a} + \sqrt{b} \sqrt{b}}{2 a - b}$

Step 1.16.1.1.2

Raise $\sqrt{2 a}$ to the power of $1$ .

$\frac{2 a - 2 \sqrt{2 a b} + b}{2 a - b} - \frac{{\sqrt{2 a}}^{1} {\sqrt{2 a}}^{1} + \sqrt{2 a} \sqrt{b} + \sqrt{b} \sqrt{2 a} + \sqrt{b} \sqrt{b}}{2 a - b}$

Step 1.16.1.1.3

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

$\frac{2 a - 2 \sqrt{2 a b} + b}{2 a - b} - \frac{{\sqrt{2 a}}^{1 + 1} + \sqrt{2 a} \sqrt{b} + \sqrt{b} \sqrt{2 a} + \sqrt{b} \sqrt{b}}{2 a - b}$

Step 1.16.1.1.4

Add $1$ and $1$ .

$\frac{2 a - 2 \sqrt{2 a b} + b}{2 a - b} - \frac{{\sqrt{2 a}}^{2} + \sqrt{2 a} \sqrt{b} + \sqrt{b} \sqrt{2 a} + \sqrt{b} \sqrt{b}}{2 a - b}$

Step 1.16.1.2

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

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Step 1.16.1.2.1

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

$\frac{2 a - 2 \sqrt{2 a b} + b}{2 a - b} - \frac{{({(2 a)}^{\frac{1}{2}})}^{2} + \sqrt{2 a} \sqrt{b} + \sqrt{b} \sqrt{2 a} + \sqrt{b} \sqrt{b}}{2 a - b}$

Step 1.16.1.2.2

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

$\frac{2 a - 2 \sqrt{2 a b} + b}{2 a - b} - \frac{{(2 a)}^{\frac{1}{2} \cdot 2} + \sqrt{2 a} \sqrt{b} + \sqrt{b} \sqrt{2 a} + \sqrt{b} \sqrt{b}}{2 a - b}$

Step 1.16.1.2.3

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

$\frac{2 a - 2 \sqrt{2 a b} + b}{2 a - b} - \frac{{(2 a)}^{\frac{2}{2}} + \sqrt{2 a} \sqrt{b} + \sqrt{b} \sqrt{2 a} + \sqrt{b} \sqrt{b}}{2 a - b}$

Step 1.16.1.2.4

Cancel the common factor of $2$ .

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Step 1.16.1.2.4.1

Cancel the common factor.

$\frac{2 a - 2 \sqrt{2 a b} + b}{2 a - b} - \frac{{(2 a)}^{\frac{2}{2}} + \sqrt{2 a} \sqrt{b} + \sqrt{b} \sqrt{2 a} + \sqrt{b} \sqrt{b}}{2 a - b}$

Step 1.16.1.2.4.2

Rewrite the expression.

$\frac{2 a - 2 \sqrt{2 a b} + b}{2 a - b} - \frac{{(2 a)}^{1} + \sqrt{2 a} \sqrt{b} + \sqrt{b} \sqrt{2 a} + \sqrt{b} \sqrt{b}}{2 a - b}$

Step 1.16.1.2.5

Simplify.

$\frac{2 a - 2 \sqrt{2 a b} + b}{2 a - b} - \frac{2 a + \sqrt{2 a} \sqrt{b} + \sqrt{b} \sqrt{2 a} + \sqrt{b} \sqrt{b}}{2 a - b}$

Step 1.16.1.3

Combine using the product rule for radicals.

$\frac{2 a - 2 \sqrt{2 a b} + b}{2 a - b} - \frac{2 a + \sqrt{2 a b} + \sqrt{b} \sqrt{2 a} + \sqrt{b} \sqrt{b}}{2 a - b}$

Step 1.16.1.4

Combine using the product rule for radicals.

$\frac{2 a - 2 \sqrt{2 a b} + b}{2 a - b} - \frac{2 a + \sqrt{2 a b} + \sqrt{b \cdot 2 a} + \sqrt{b} \sqrt{b}}{2 a - b}$

Step 1.16.1.5

Multiply $\sqrt{b} \sqrt{b}$ .

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Step 1.16.1.5.1

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

$\frac{2 a - 2 \sqrt{2 a b} + b}{2 a - b} - \frac{2 a + \sqrt{2 a b} + \sqrt{b \cdot 2 a} + {\sqrt{b}}^{1} \sqrt{b}}{2 a - b}$

Step 1.16.1.5.2

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

$\frac{2 a - 2 \sqrt{2 a b} + b}{2 a - b} - \frac{2 a + \sqrt{2 a b} + \sqrt{b \cdot 2 a} + {\sqrt{b}}^{1} {\sqrt{b}}^{1}}{2 a - b}$

Step 1.16.1.5.3

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

$\frac{2 a - 2 \sqrt{2 a b} + b}{2 a - b} - \frac{2 a + \sqrt{2 a b} + \sqrt{b \cdot 2 a} + {\sqrt{b}}^{1 + 1}}{2 a - b}$

Step 1.16.1.5.4

Add $1$ and $1$ .

$\frac{2 a - 2 \sqrt{2 a b} + b}{2 a - b} - \frac{2 a + \sqrt{2 a b} + \sqrt{b \cdot 2 a} + {\sqrt{b}}^{2}}{2 a - b}$

Step 1.16.1.6

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

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Step 1.16.1.6.1

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

$\frac{2 a - 2 \sqrt{2 a b} + b}{2 a - b} - \frac{2 a + \sqrt{2 a b} + \sqrt{b \cdot 2 a} + {(b^{\frac{1}{2}})}^{2}}{2 a - b}$

Step 1.16.1.6.2

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

$\frac{2 a - 2 \sqrt{2 a b} + b}{2 a - b} - \frac{2 a + \sqrt{2 a b} + \sqrt{b \cdot 2 a} + b^{\frac{1}{2} \cdot 2}}{2 a - b}$

Step 1.16.1.6.3

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

$\frac{2 a - 2 \sqrt{2 a b} + b}{2 a - b} - \frac{2 a + \sqrt{2 a b} + \sqrt{b \cdot 2 a} + b^{\frac{2}{2}}}{2 a - b}$

Step 1.16.1.6.4

Cancel the common factor of $2$ .

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Step 1.16.1.6.4.1

Cancel the common factor.

$\frac{2 a - 2 \sqrt{2 a b} + b}{2 a - b} - \frac{2 a + \sqrt{2 a b} + \sqrt{b \cdot 2 a} + b^{\frac{2}{2}}}{2 a - b}$

Step 1.16.1.6.4.2

Rewrite the expression.

$\frac{2 a - 2 \sqrt{2 a b} + b}{2 a - b} - \frac{2 a + \sqrt{2 a b} + \sqrt{b \cdot 2 a} + b^{1}}{2 a - b}$

Step 1.16.1.6.5

Simplify.

$\frac{2 a - 2 \sqrt{2 a b} + b}{2 a - b} - \frac{2 a + \sqrt{2 a b} + \sqrt{b \cdot 2 a} + b}{2 a - b}$

Step 1.16.2

Add $\sqrt{2 a b}$ and $\sqrt{b \cdot 2 a}$ .

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Step 1.16.2.1

Move $b$ .

$\frac{2 a - 2 \sqrt{2 a b} + b}{2 a - b} - \frac{2 a + \sqrt{2 a b} + \sqrt{2 a b} + b}{2 a - b}$

Step 1.16.2.2

Add $\sqrt{2 a b}$ and $\sqrt{2 a b}$ .

$\frac{2 a - 2 \sqrt{2 a b} + b}{2 a - b} - \frac{2 a + 2 \sqrt{2 a b} + b}{2 a - b}$

Step 2

Combine the numerators over the common denominator.

$\frac{2 a - 2 \sqrt{2 a b} + b - (2 a + 2 \sqrt{2 a b} + b)}{2 a - b}$

Step 3

Simplify each term.

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Step 3.1

Apply the distributive property.

$\frac{2 a - 2 \sqrt{2 a b} + b - (2 a) - (2 \sqrt{2 a b}) - b}{2 a - b}$

Step 3.2

Simplify.

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Step 3.2.1

Multiply $2$ by $- 1$ .

$\frac{2 a - 2 \sqrt{2 a b} + b - 2 a - (2 \sqrt{2 a b}) - b}{2 a - b}$

Step 3.2.2

Multiply $2$ by $- 1$ .

$\frac{2 a - 2 \sqrt{2 a b} + b - 2 a - 2 \sqrt{2 a b} - b}{2 a - b}$

Step 4

Simplify by adding terms.

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Step 4.1

Combine the opposite terms in $2 a - 2 \sqrt{2 a b} + b - 2 a - 2 \sqrt{2 a b} - b$ .

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Step 4.1.1

Subtract $2 a$ from $2 a$ .

$\frac{0 - 2 \sqrt{2 a b} + b - 2 \sqrt{2 a b} - b}{2 a - b}$

Step 4.1.2

Subtract $2 \sqrt{2 a b}$ from $0$ .

$\frac{- 2 \sqrt{2 a b} + b - 2 \sqrt{2 a b} - b}{2 a - b}$

Step 4.1.3

Subtract $b$ from $b$ .

$\frac{- 2 \sqrt{2 a b} + 0 - 2 \sqrt{2 a b}}{2 a - b}$

Step 4.1.4

Add $- 2 \sqrt{2 a b}$ and $0$ .

$\frac{- 2 \sqrt{2 a b} - 2 \sqrt{2 a b}}{2 a - b}$

Step 4.2

Subtract $2 \sqrt{2 a b}$ from $- 2 \sqrt{2 a b}$ .

$\frac{- 4 \sqrt{2 a b}}{2 a - b}$

Step 4.3

Move the negative in front of the fraction.

$- \frac{4 \sqrt{2 a b}}{2 a - b}$