Calculus Examples

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

Step 1

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

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

Step 2

Combine fractions.

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Step 2.1

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

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

Step 2.2

Expand the denominator using the FOIL method.

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

Step 2.3

Simplify.

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

Step 3

Simplify the numerator.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

Add $1$ and $1$ .

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

Step 4

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

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

Step 5

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

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Step 5.1

Apply the distributive property.

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

Step 5.2

Apply the distributive property.

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

Step 5.3

Apply the distributive property.

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

Step 6

Simplify and combine like terms.

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Step 6.1

Simplify each term.

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Step 6.1.1

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

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Step 6.1.1.1

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

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

Step 6.1.1.2

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

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

Step 6.1.1.3

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

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

Step 6.1.1.4

Add $1$ and $1$ .

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

Step 6.1.2

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

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Step 6.1.2.1

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

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

Step 6.1.2.2

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

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

Step 6.1.2.3

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

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

Step 6.1.2.4

Cancel the common factor of $2$ .

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Step 6.1.2.4.1

Cancel the common factor.

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

Step 6.1.2.4.2

Rewrite the expression.

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

Step 6.1.2.5

Simplify.

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

Step 6.1.3

Combine using the product rule for radicals.

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

Step 6.1.4

Combine using the product rule for radicals.

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

Step 6.1.5

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

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Step 6.1.5.1

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

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

Step 6.1.5.2

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

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

Step 6.1.5.3

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

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

Step 6.1.5.4

Add $1$ and $1$ .

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

Step 6.1.6

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

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Step 6.1.6.1

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

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

Step 6.1.6.2

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

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

Step 6.1.6.3

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

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

Step 6.1.6.4

Cancel the common factor of $2$ .

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Step 6.1.6.4.1

Cancel the common factor.

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

Step 6.1.6.4.2

Rewrite the expression.

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

Step 6.1.6.5

Simplify.

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

Step 6.2

Add $a$ and $a$ .

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

Step 6.3

Subtract $b$ from $b$ .

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

Step 6.4

Add $2 a$ and $0$ .

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

Step 7

Simplify the numerator.

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Step 7.1

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

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

Step 7.2

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

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

Step 7.3

Reorder the factors of $(a - b) (a + b)$ .

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

Step 7.4

Add ${((a + b) (a - b))}^{\frac{1}{2}}$ and ${((a + b) (a - b))}^{\frac{1}{2}}$ .

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

Step 7.5

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

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Step 7.5.1

Factor $2$ out of $2 a$ .

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

Step 7.5.2

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

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

Step 7.5.3

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

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

Step 7.6

Expand $(a + b) (a - b)$ using the FOIL Method.

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Step 7.6.1

Apply the distributive property.

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

Step 7.6.2

Apply the distributive property.

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

Step 7.6.3

Apply the distributive property.

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

Step 7.7

Combine the opposite terms in $a \cdot a + a (- b) + b a + b (- b)$ .

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Step 7.7.1

Reorder the factors in the terms $a (- b)$ and $b a$ .

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

Step 7.7.2

Add $- a b$ and $a b$ .

$\frac{2 (a + {(a \cdot a + 0 + b (- b))}^{\frac{1}{2}})}{2 b}$

Step 7.7.3

Add $a \cdot a$ and $0$ .

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

Step 7.8

Simplify each term.

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Step 7.8.1

Multiply $a$ by $a$ .

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

Step 7.8.2

Rewrite using the commutative property of multiplication.

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

Step 7.8.3

Multiply $b$ by $b$ by adding the exponents.

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Step 7.8.3.1

Move $b$ .

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

Step 7.8.3.2

Multiply $b$ by $b$ .

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

Step 8

Reduce the expression by cancelling the common factors.

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Step 8.1

Cancel the common factor.

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

Step 8.2

Rewrite the expression.

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