Algebra Examples

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

Step 1

To write $\frac{a - b}{a + b}$ as a fraction with a common denominator, multiply by $\frac{a - b}{a - b}$ .

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

Step 2

To write $\frac{a + b}{a - b}$ as a fraction with a common denominator, multiply by $\frac{a + b}{a + b}$ .

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

Step 3

Write each expression with a common denominator of $(a + b) (a - b)$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

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Step 5.1

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

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Step 5.1.1

Apply the distributive property.

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

Step 5.1.2

Apply the distributive property.

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

Step 5.1.3

Apply the distributive property.

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

Step 5.2

Simplify and combine like terms.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

Multiply $a$ by $a$ .

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

Step 5.2.1.2

Rewrite using the commutative property of multiplication.

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

Step 5.2.1.3

Rewrite using the commutative property of multiplication.

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

Step 5.2.1.4

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

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Step 5.2.1.4.1

Move $b$ .

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

Step 5.2.1.4.2

Multiply $b$ by $b$ .

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

Step 5.2.1.5

Multiply $- 1$ by $- 1$ .

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

Step 5.2.1.6

Multiply $b^{2}$ by $1$ .

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

Step 5.2.2

Subtract $b a$ from $- a b$ .

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Step 5.2.2.1

Move $b$ .

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

Step 5.2.2.2

Subtract $a b$ from $- a b$ .

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

Step 5.3

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

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Step 5.3.1

Apply the distributive property.

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

Step 5.3.2

Apply the distributive property.

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

Step 5.3.3

Apply the distributive property.

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

Step 5.4

Simplify and combine like terms.

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Step 5.4.1

Simplify each term.

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Step 5.4.1.1

Multiply $a$ by $a$ .

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

Step 5.4.1.2

Multiply $b$ by $b$ .

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

Step 5.4.2

Add $a b$ and $b a$ .

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Step 5.4.2.1

Reorder $b$ and $a$ .

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

Step 5.4.2.2

Add $a b$ and $a b$ .

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

Step 5.5

Add $a^{2}$ and $a^{2}$ .

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

Step 5.6

Add $- 2 a b$ and $2 a b$ .

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

Step 5.7

Add $2 a^{2}$ and $0$ .

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

Step 5.8

Add $b^{2}$ and $b^{2}$ .

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

Step 5.9

Factor $2$ out of $2 a^{2} + 2 b^{2}$ .

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Step 5.9.1

Factor $2$ out of $2 a^{2}$ .

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

Step 5.9.2

Factor $2$ out of $2 b^{2}$ .

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

Step 5.9.3

Factor $2$ out of $2 (a^{2}) + 2 (b^{2})$ .

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

Step 6

Combine into one fraction.

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Step 6.1

Write $1$ as a fraction with a common denominator.

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

Step 6.2

Combine the numerators over the common denominator.

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

Step 7

Factor using the perfect square rule.

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Step 7.1

Rearrange terms.

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

Step 7.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 a b = 2 \cdot a \cdot b$

Step 7.3

Rewrite the polynomial.

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

Step 7.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = a$ and $b = b$ .

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

Step 8

Simplify terms.

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Step 8.1

Combine.

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

Step 8.2

Cancel the common factor of $a^{2} + b^{2}$ .

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Step 8.2.1

Factor $a^{2} + b^{2}$ out of $2 (a^{2} + b^{2}) {(a + b)}^{2}$ .

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

Step 8.2.2

Cancel the common factor.

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

Step 8.2.3

Rewrite the expression.

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

Step 8.3

Cancel the common factor of $a b$ .

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Step 8.3.1

Factor $a b$ out of $(a + b) (a - b) \cdot 2 a b$ .

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

Step 8.3.2

Cancel the common factor.

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

Step 8.3.3

Rewrite the expression.

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

Step 8.4

Cancel the common factor of $2$ .

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Step 8.4.1

Cancel the common factor.

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

Step 8.4.2

Rewrite the expression.

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

Step 8.5

Cancel the common factor of ${(a + b)}^{2}$ and $a + b$ .

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Step 8.5.1

Factor $a + b$ out of ${(a + b)}^{2}$ .

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

Step 8.5.2

Cancel the common factors.

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Step 8.5.2.1

Cancel the common factor.

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

Step 8.5.2.2

Rewrite the expression.

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