Basic Math Examples

$(\frac{{(a + b)}^{2}}{4 a b} - 1) (\frac{{(a - b)}^{2}}{4 a b} + 1)$

Step 1

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{4 a b}{4 a b}$ .

$(\frac{{(a + b)}^{2}}{4 a b} - 1 \cdot \frac{4 a b}{4 a b}) (\frac{{(a - b)}^{2}}{4 a b} + 1)$

Step 2

Simplify terms.

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Step 2.1

Combine $- 1$ and $\frac{4 a b}{4 a b}$ .

$(\frac{{(a + b)}^{2}}{4 a b} + \frac{- (4 a b)}{4 a b}) (\frac{{(a - b)}^{2}}{4 a b} + 1)$

Step 2.2

Combine the numerators over the common denominator.

$\frac{{(a + b)}^{2} - (4 a b)}{4 a b} (\frac{{(a - b)}^{2}}{4 a b} + 1)$

Step 3

Simplify the numerator.

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Step 3.1

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

$\frac{(a + b) (a + b) - (4 a b)}{4 a b} (\frac{{(a - b)}^{2}}{4 a b} + 1)$

Step 3.2

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

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Step 3.2.1

Apply the distributive property.

$\frac{a (a + b) + b (a + b) - (4 a b)}{4 a b} (\frac{{(a - b)}^{2}}{4 a b} + 1)$

Step 3.2.2

Apply the distributive property.

$\frac{a \cdot a + a b + b (a + b) - (4 a b)}{4 a b} (\frac{{(a - b)}^{2}}{4 a b} + 1)$

Step 3.2.3

Apply the distributive property.

$\frac{a \cdot a + a b + b a + b \cdot b - (4 a b)}{4 a b} (\frac{{(a - b)}^{2}}{4 a b} + 1)$

Step 3.3

Simplify and combine like terms.

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Step 3.3.1

Simplify each term.

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Step 3.3.1.1

Multiply $a$ by $a$ .

$\frac{a^{2} + a b + b a + b \cdot b - (4 a b)}{4 a b} (\frac{{(a - b)}^{2}}{4 a b} + 1)$

Step 3.3.1.2

Multiply $b$ by $b$ .

$\frac{a^{2} + a b + b a + b^{2} - (4 a b)}{4 a b} (\frac{{(a - b)}^{2}}{4 a b} + 1)$

Step 3.3.2

Add $a b$ and $b a$ .

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Step 3.3.2.1

Reorder $b$ and $a$ .

$\frac{a^{2} + a b + a b + b^{2} - (4 a b)}{4 a b} (\frac{{(a - b)}^{2}}{4 a b} + 1)$

Step 3.3.2.2

Add $a b$ and $a b$ .

$\frac{a^{2} + 2 a b + b^{2} - (4 a b)}{4 a b} (\frac{{(a - b)}^{2}}{4 a b} + 1)$

Step 3.4

Multiply $4$ by $- 1$ .

$\frac{a^{2} + 2 a b + b^{2} - 4 (a b)}{4 a b} (\frac{{(a - b)}^{2}}{4 a b} + 1)$

Step 3.5

Subtract $4 a b$ from $2 a b$ .

$\frac{a^{2} + b^{2} - 2 a b}{4 a b} (\frac{{(a - b)}^{2}}{4 a b} + 1)$

Step 3.6

Factor using the perfect square rule.

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Step 3.6.1

Rearrange terms.

$\frac{a^{2} - 2 a b + b^{2}}{4 a b} (\frac{{(a - b)}^{2}}{4 a b} + 1)$

Step 3.6.2

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 3.6.3

Rewrite the polynomial.

$\frac{a^{2} - 2 \cdot a \cdot b + b^{2}}{4 a b} (\frac{{(a - b)}^{2}}{4 a b} + 1)$

Step 3.6.4

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

$\frac{{(a - b)}^{2}}{4 a b} (\frac{{(a - b)}^{2}}{4 a b} + 1)$

Step 4

Combine into one fraction.

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Step 4.1

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

$\frac{{(a - b)}^{2}}{4 a b} (\frac{{(a - b)}^{2}}{4 a b} + \frac{4 a b}{4 a b})$

Step 4.2

Combine the numerators over the common denominator.

$\frac{{(a - b)}^{2}}{4 a b} \cdot \frac{{(a - b)}^{2} + 4 a b}{4 a b}$

Step 5

Simplify the numerator.

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Step 5.1

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

$\frac{{(a - b)}^{2}}{4 a b} \cdot \frac{(a - b) (a - b) + 4 a b}{4 a b}$

Step 5.2

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

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Step 5.2.1

Apply the distributive property.

$\frac{{(a - b)}^{2}}{4 a b} \cdot \frac{a (a - b) - b (a - b) + 4 a b}{4 a b}$

Step 5.2.2

Apply the distributive property.

$\frac{{(a - b)}^{2}}{4 a b} \cdot \frac{a \cdot a + a (- b) - b (a - b) + 4 a b}{4 a b}$

Step 5.2.3

Apply the distributive property.

$\frac{{(a - b)}^{2}}{4 a b} \cdot \frac{a \cdot a + a (- b) - b a - b (- b) + 4 a b}{4 a b}$

Step 5.3

Simplify and combine like terms.

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Step 5.3.1

Simplify each term.

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Step 5.3.1.1

Multiply $a$ by $a$ .

$\frac{{(a - b)}^{2}}{4 a b} \cdot \frac{a^{2} + a (- b) - b a - b (- b) + 4 a b}{4 a b}$

Step 5.3.1.2

Rewrite using the commutative property of multiplication.

$\frac{{(a - b)}^{2}}{4 a b} \cdot \frac{a^{2} - a b - b a - b (- b) + 4 a b}{4 a b}$

Step 5.3.1.3

Rewrite using the commutative property of multiplication.

$\frac{{(a - b)}^{2}}{4 a b} \cdot \frac{a^{2} - a b - b a - 1 \cdot - 1 b \cdot b + 4 a b}{4 a b}$

Step 5.3.1.4

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

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Step 5.3.1.4.1

Move $b$ .

$\frac{{(a - b)}^{2}}{4 a b} \cdot \frac{a^{2} - a b - b a - 1 \cdot - 1 (b \cdot b) + 4 a b}{4 a b}$

Step 5.3.1.4.2

Multiply $b$ by $b$ .

$\frac{{(a - b)}^{2}}{4 a b} \cdot \frac{a^{2} - a b - b a - 1 \cdot - 1 b^{2} + 4 a b}{4 a b}$

Step 5.3.1.5

Multiply $- 1$ by $- 1$ .

$\frac{{(a - b)}^{2}}{4 a b} \cdot \frac{a^{2} - a b - b a + 1 b^{2} + 4 a b}{4 a b}$

Step 5.3.1.6

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

$\frac{{(a - b)}^{2}}{4 a b} \cdot \frac{a^{2} - a b - b a + b^{2} + 4 a b}{4 a b}$

Step 5.3.2

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

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Step 5.3.2.1

Move $b$ .

$\frac{{(a - b)}^{2}}{4 a b} \cdot \frac{a^{2} - a b - 1 a b + b^{2} + 4 a b}{4 a b}$

Step 5.3.2.2

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

$\frac{{(a - b)}^{2}}{4 a b} \cdot \frac{a^{2} - 2 a b + b^{2} + 4 a b}{4 a b}$

Step 5.4

Add $- 2 a b$ and $4 a b$ .

$\frac{{(a - b)}^{2}}{4 a b} \cdot \frac{a^{2} + b^{2} + 2 a b}{4 a b}$

Step 5.5

Factor using the perfect square rule.

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Step 5.5.1

Rearrange terms.

$\frac{{(a - b)}^{2}}{4 a b} \cdot \frac{a^{2} + 2 a b + b^{2}}{4 a b}$

Step 5.5.2

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 5.5.3

Rewrite the polynomial.

$\frac{{(a - b)}^{2}}{4 a b} \cdot \frac{a^{2} + 2 \cdot a \cdot b + b^{2}}{4 a b}$

Step 5.5.4

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

$\frac{{(a - b)}^{2}}{4 a b} \cdot \frac{{(a + b)}^{2}}{4 a b}$

Step 6

Combine.

$\frac{{(a - b)}^{2} {(a + b)}^{2}}{4 a b (4 a b)}$

Step 7

Multiply $a$ by $a$ by adding the exponents.

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Step 7.1

Move $a$ .

$\frac{{(a - b)}^{2} {(a + b)}^{2}}{4 (a \cdot a) b (4 b)}$

Step 7.2

Multiply $a$ by $a$ .

$\frac{{(a - b)}^{2} {(a + b)}^{2}}{4 a^{2} b (4 b)}$

Step 8

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

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Step 8.1

Move $b$ .

$\frac{{(a - b)}^{2} {(a + b)}^{2}}{4 a^{2} (b \cdot b) \cdot 4}$

Step 8.2

Multiply $b$ by $b$ .

$\frac{{(a - b)}^{2} {(a + b)}^{2}}{4 a^{2} b^{2} \cdot 4}$

Step 9

Multiply $4$ by $4$ .

$\frac{{(a - b)}^{2} {(a + b)}^{2}}{16 a^{2} b^{2}}$