Basic Math Examples

$\frac{{(a + b)}^{3} + {(a - b)}^{3}}{a (a^{2} + 3 b^{2})}$

Step 1

Simplify the numerator.

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Step 1.1

Since both terms are perfect cubes, factor using the sum of cubes formula, $a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$ where $a = a + b$ and $b = a - b$ .

$\frac{(a + b + a - b) ({(a + b)}^{2} - (a + b) (a - b) + {(a - b)}^{2})}{a (a^{2} + 3 b^{2})}$

Step 1.2

Simplify.

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Step 1.2.1

Add $a$ and $a$ .

$\frac{(2 a + b - b) ({(a + b)}^{2} - (a + b) (a - b) + {(a - b)}^{2})}{a (a^{2} + 3 b^{2})}$

Step 1.2.2

Subtract $b$ from $b$ .

$\frac{(2 a + 0) ({(a + b)}^{2} - (a + b) (a - b) + {(a - b)}^{2})}{a (a^{2} + 3 b^{2})}$

Step 1.2.3

Add $2 a$ and $0$ .

$\frac{2 a ({(a + b)}^{2} - (a + b) (a - b) + {(a - b)}^{2})}{a (a^{2} + 3 b^{2})}$

Step 1.2.4

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

$\frac{2 a ((a + b) (a + b) - (a + b) (a - b) + {(a - b)}^{2})}{a (a^{2} + 3 b^{2})}$

Step 1.2.5

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

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Step 1.2.5.1

Apply the distributive property.

$\frac{2 a (a (a + b) + b (a + b) - (a + b) (a - b) + {(a - b)}^{2})}{a (a^{2} + 3 b^{2})}$

Step 1.2.5.2

Apply the distributive property.

$\frac{2 a (a \cdot a + a b + b (a + b) - (a + b) (a - b) + {(a - b)}^{2})}{a (a^{2} + 3 b^{2})}$

Step 1.2.5.3

Apply the distributive property.

$\frac{2 a (a \cdot a + a b + b a + b \cdot b - (a + b) (a - b) + {(a - b)}^{2})}{a (a^{2} + 3 b^{2})}$

Step 1.2.6

Simplify and combine like terms.

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Step 1.2.6.1

Simplify each term.

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Step 1.2.6.1.1

Multiply $a$ by $a$ .

$\frac{2 a (a^{2} + a b + b a + b \cdot b - (a + b) (a - b) + {(a - b)}^{2})}{a (a^{2} + 3 b^{2})}$

Step 1.2.6.1.2

Multiply $b$ by $b$ .

$\frac{2 a (a^{2} + a b + b a + b^{2} - (a + b) (a - b) + {(a - b)}^{2})}{a (a^{2} + 3 b^{2})}$

Step 1.2.6.2

Add $a b$ and $b a$ .

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Step 1.2.6.2.1

Reorder $b$ and $a$ .

$\frac{2 a (a^{2} + a b + a b + b^{2} - (a + b) (a - b) + {(a - b)}^{2})}{a (a^{2} + 3 b^{2})}$

Step 1.2.6.2.2

Add $a b$ and $a b$ .

$\frac{2 a (a^{2} + 2 a b + b^{2} - (a + b) (a - b) + {(a - b)}^{2})}{a (a^{2} + 3 b^{2})}$

Step 1.2.7

Apply the distributive property.

$\frac{2 a (a^{2} + 2 a b + b^{2} + (- a - b) (a - b) + {(a - b)}^{2})}{a (a^{2} + 3 b^{2})}$

Step 1.2.8

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

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Step 1.2.8.1

Apply the distributive property.

$\frac{2 a (a^{2} + 2 a b + b^{2} - a (a - b) - b (a - b) + {(a - b)}^{2})}{a (a^{2} + 3 b^{2})}$

Step 1.2.8.2

Apply the distributive property.

$\frac{2 a (a^{2} + 2 a b + b^{2} - a \cdot a - a (- b) - b (a - b) + {(a - b)}^{2})}{a (a^{2} + 3 b^{2})}$

Step 1.2.8.3

Apply the distributive property.

$\frac{2 a (a^{2} + 2 a b + b^{2} - a \cdot a - a (- b) - b a - b (- b) + {(a - b)}^{2})}{a (a^{2} + 3 b^{2})}$

Step 1.2.9

Simplify and combine like terms.

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Step 1.2.9.1

Simplify each term.

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Step 1.2.9.1.1

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

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Step 1.2.9.1.1.1

Move $a$ .

$\frac{2 a (a^{2} + 2 a b + b^{2} - (a \cdot a) - a (- b) - b a - b (- b) + {(a - b)}^{2})}{a (a^{2} + 3 b^{2})}$

Step 1.2.9.1.1.2

Multiply $a$ by $a$ .

$\frac{2 a (a^{2} + 2 a b + b^{2} - a^{2} - a (- b) - b a - b (- b) + {(a - b)}^{2})}{a (a^{2} + 3 b^{2})}$

Step 1.2.9.1.2

Rewrite using the commutative property of multiplication.

$\frac{2 a (a^{2} + 2 a b + b^{2} - a^{2} - 1 \cdot - 1 a b - b a - b (- b) + {(a - b)}^{2})}{a (a^{2} + 3 b^{2})}$

Step 1.2.9.1.3

Multiply $- 1$ by $- 1$ .

$\frac{2 a (a^{2} + 2 a b + b^{2} - a^{2} + 1 a b - b a - b (- b) + {(a - b)}^{2})}{a (a^{2} + 3 b^{2})}$

Step 1.2.9.1.4

Multiply $a$ by $1$ .

$\frac{2 a (a^{2} + 2 a b + b^{2} - a^{2} + a b - b a - b (- b) + {(a - b)}^{2})}{a (a^{2} + 3 b^{2})}$

Step 1.2.9.1.5

Rewrite using the commutative property of multiplication.

$\frac{2 a (a^{2} + 2 a b + b^{2} - a^{2} + a b - b a - 1 \cdot - 1 b \cdot b + {(a - b)}^{2})}{a (a^{2} + 3 b^{2})}$

Step 1.2.9.1.6

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

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Step 1.2.9.1.6.1

Move $b$ .

$\frac{2 a (a^{2} + 2 a b + b^{2} - a^{2} + a b - b a - 1 \cdot - 1 (b \cdot b) + {(a - b)}^{2})}{a (a^{2} + 3 b^{2})}$

Step 1.2.9.1.6.2

Multiply $b$ by $b$ .

$\frac{2 a (a^{2} + 2 a b + b^{2} - a^{2} + a b - b a - 1 \cdot - 1 b^{2} + {(a - b)}^{2})}{a (a^{2} + 3 b^{2})}$

Step 1.2.9.1.7

Multiply $- 1$ by $- 1$ .

$\frac{2 a (a^{2} + 2 a b + b^{2} - a^{2} + a b - b a + 1 b^{2} + {(a - b)}^{2})}{a (a^{2} + 3 b^{2})}$

Step 1.2.9.1.8

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

$\frac{2 a (a^{2} + 2 a b + b^{2} - a^{2} + a b - b a + b^{2} + {(a - b)}^{2})}{a (a^{2} + 3 b^{2})}$

Step 1.2.9.2

Subtract $b a$ from $a b$ .

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Step 1.2.9.2.1

Move $b$ .

$\frac{2 a (a^{2} + 2 a b + b^{2} - a^{2} + a b - 1 a b + b^{2} + {(a - b)}^{2})}{a (a^{2} + 3 b^{2})}$

Step 1.2.9.2.2

Subtract $a b$ from $a b$ .

$\frac{2 a (a^{2} + 2 a b + b^{2} - a^{2} + 0 + b^{2} + {(a - b)}^{2})}{a (a^{2} + 3 b^{2})}$

Step 1.2.9.3

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

$\frac{2 a (a^{2} + 2 a b + b^{2} - a^{2} + b^{2} + {(a - b)}^{2})}{a (a^{2} + 3 b^{2})}$

Step 1.2.10

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

$\frac{2 a (a^{2} + 2 a b + b^{2} - a^{2} + b^{2} + (a - b) (a - b))}{a (a^{2} + 3 b^{2})}$

Step 1.2.11

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

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Step 1.2.11.1

Apply the distributive property.

$\frac{2 a (a^{2} + 2 a b + b^{2} - a^{2} + b^{2} + a (a - b) - b (a - b))}{a (a^{2} + 3 b^{2})}$

Step 1.2.11.2

Apply the distributive property.

$\frac{2 a (a^{2} + 2 a b + b^{2} - a^{2} + b^{2} + a \cdot a + a (- b) - b (a - b))}{a (a^{2} + 3 b^{2})}$

Step 1.2.11.3

Apply the distributive property.

$\frac{2 a (a^{2} + 2 a b + b^{2} - a^{2} + b^{2} + a \cdot a + a (- b) - b a - b (- b))}{a (a^{2} + 3 b^{2})}$

Step 1.2.12

Simplify and combine like terms.

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Step 1.2.12.1

Simplify each term.

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Step 1.2.12.1.1

Multiply $a$ by $a$ .

$\frac{2 a (a^{2} + 2 a b + b^{2} - a^{2} + b^{2} + a^{2} + a (- b) - b a - b (- b))}{a (a^{2} + 3 b^{2})}$

Step 1.2.12.1.2

Rewrite using the commutative property of multiplication.

$\frac{2 a (a^{2} + 2 a b + b^{2} - a^{2} + b^{2} + a^{2} - a b - b a - b (- b))}{a (a^{2} + 3 b^{2})}$

Step 1.2.12.1.3

Rewrite using the commutative property of multiplication.

$\frac{2 a (a^{2} + 2 a b + b^{2} - a^{2} + b^{2} + a^{2} - a b - b a - 1 \cdot - 1 b \cdot b)}{a (a^{2} + 3 b^{2})}$

Step 1.2.12.1.4

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

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Step 1.2.12.1.4.1

Move $b$ .

$\frac{2 a (a^{2} + 2 a b + b^{2} - a^{2} + b^{2} + a^{2} - a b - b a - 1 \cdot - 1 (b \cdot b))}{a (a^{2} + 3 b^{2})}$

Step 1.2.12.1.4.2

Multiply $b$ by $b$ .

$\frac{2 a (a^{2} + 2 a b + b^{2} - a^{2} + b^{2} + a^{2} - a b - b a - 1 \cdot - 1 b^{2})}{a (a^{2} + 3 b^{2})}$

Step 1.2.12.1.5

Multiply $- 1$ by $- 1$ .

$\frac{2 a (a^{2} + 2 a b + b^{2} - a^{2} + b^{2} + a^{2} - a b - b a + 1 b^{2})}{a (a^{2} + 3 b^{2})}$

Step 1.2.12.1.6

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

$\frac{2 a (a^{2} + 2 a b + b^{2} - a^{2} + b^{2} + a^{2} - a b - b a + b^{2})}{a (a^{2} + 3 b^{2})}$

Step 1.2.12.2

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

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Step 1.2.12.2.1

Move $b$ .

$\frac{2 a (a^{2} + 2 a b + b^{2} - a^{2} + b^{2} + a^{2} - a b - 1 a b + b^{2})}{a (a^{2} + 3 b^{2})}$

Step 1.2.12.2.2

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

$\frac{2 a (a^{2} + 2 a b + b^{2} - a^{2} + b^{2} + a^{2} - 2 a b + b^{2})}{a (a^{2} + 3 b^{2})}$

Step 1.2.13

Subtract $a^{2}$ from $a^{2}$ .

$\frac{2 a (2 a b + b^{2} + 0 + b^{2} + a^{2} - 2 a b + b^{2})}{a (a^{2} + 3 b^{2})}$

Step 1.2.14

Add $2 a b$ and $0$ .

$\frac{2 a (b^{2} + 2 a b + b^{2} + a^{2} - 2 a b + b^{2})}{a (a^{2} + 3 b^{2})}$

Step 1.2.15

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

$\frac{2 a (2 b^{2} + 2 a b + a^{2} - 2 a b + b^{2})}{a (a^{2} + 3 b^{2})}$

Step 1.2.16

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

$\frac{2 a (3 b^{2} + 2 a b + a^{2} - 2 a b)}{a (a^{2} + 3 b^{2})}$

Step 1.2.17

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

$\frac{2 a (3 b^{2} + a^{2} + 0)}{a (a^{2} + 3 b^{2})}$

Step 1.2.18

Add $3 b^{2} + a^{2}$ and $0$ .

$\frac{2 a (3 b^{2} + a^{2})}{a (a^{2} + 3 b^{2})}$

Step 2

Reduce the expression by cancelling the common factors.

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Step 2.1

Cancel the common factor of $a$ .

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Step 2.1.1

Cancel the common factor.

$\frac{2 a (3 b^{2} + a^{2})}{a (a^{2} + 3 b^{2})}$

Step 2.1.2

Rewrite the expression.

$\frac{2 (3 b^{2} + a^{2})}{a^{2} + 3 b^{2}}$

Step 2.2

Cancel the common factor of $3 b^{2} + a^{2}$ and $a^{2} + 3 b^{2}$ .

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Step 2.2.1

Reorder terms.

$\frac{2 (a^{2} + 3 b^{2})}{a^{2} + 3 b^{2}}$

Step 2.2.2

Cancel the common factor.

$\frac{2 (a^{2} + 3 b^{2})}{a^{2} + 3 b^{2}}$

Step 2.2.3

Divide $2$ by $1$ .

$2$