Basic Math Examples

$4 a \div \frac{1}{a^{2} - 4} - 2 a \div \frac{7}{a^{2} - 4}$

Step 1

Simplify each term.

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Step 1.1

To divide by a fraction, multiply by its reciprocal.

$4 a (a^{2} - 4) - 2 a \div \frac{7}{a^{2} - 4}$

Step 1.2

Apply the distributive property.

$4 a \cdot a^{2} + 4 a \cdot - 4 - 2 a \div \frac{7}{a^{2} - 4}$

Step 1.3

Multiply $a$ by $a^{2}$ by adding the exponents.

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Step 1.3.1

Move $a^{2}$ .

$4 (a^{2} a) + 4 a \cdot - 4 - 2 a \div \frac{7}{a^{2} - 4}$

Step 1.3.2

Multiply $a^{2}$ by $a$ .

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Step 1.3.2.1

Raise $a$ to the power of $1$ .

$4 (a^{2} a^{1}) + 4 a \cdot - 4 - 2 a \div \frac{7}{a^{2} - 4}$

Step 1.3.2.2

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

$4 a^{2 + 1} + 4 a \cdot - 4 - 2 a \div \frac{7}{a^{2} - 4}$

Step 1.3.3

Add $2$ and $1$ .

$4 a^{3} + 4 a \cdot - 4 - 2 a \div \frac{7}{a^{2} - 4}$

Step 1.4

Multiply $- 4$ by $4$ .

$4 a^{3} - 16 a - 2 a \div \frac{7}{a^{2} - 4}$

Step 1.5

To divide by a fraction, multiply by its reciprocal.

$4 a^{3} - 16 a - (2 a \frac{a^{2} - 4}{7})$

Step 1.6

Simplify the numerator.

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Step 1.6.1

Rewrite $4$ as $2^{2}$ .

$4 a^{3} - 16 a - (2 a \frac{a^{2} - 2^{2}}{7})$

Step 1.6.2

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

$4 a^{3} - 16 a - (2 a \frac{(a + 2) (a - 2)}{7})$

Step 1.7

Multiply $2 a \frac{(a + 2) (a - 2)}{7}$ .

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Step 1.7.1

Combine $2$ and $\frac{(a + 2) (a - 2)}{7}$ .

$4 a^{3} - 16 a - (a \frac{2 ((a + 2) (a - 2))}{7})$

Step 1.7.2

Combine $a$ and $\frac{2 ((a + 2) (a - 2))}{7}$ .

$4 a^{3} - 16 a - \frac{a (2 ((a + 2) (a - 2)))}{7}$

Step 1.8

Remove unnecessary parentheses.

$4 a^{3} - 16 a - \frac{a \cdot 2 (a + 2) (a - 2)}{7}$

Step 1.9

Move $2$ to the left of $a$ .

$4 a^{3} - 16 a - \frac{2 a (a + 2) (a - 2)}{7}$

Step 2

Find the common denominator.

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Step 2.1

Write $4 a^{3}$ as a fraction with denominator $1$ .

$\frac{4 a^{3}}{1} - 16 a - \frac{2 a (a + 2) (a - 2)}{7}$

Step 2.2

Multiply $\frac{4 a^{3}}{1}$ by $\frac{7}{7}$ .

$\frac{4 a^{3}}{1} \cdot \frac{7}{7} - 16 a - \frac{2 a (a + 2) (a - 2)}{7}$

Step 2.3

Multiply $\frac{4 a^{3}}{1}$ by $\frac{7}{7}$ .

$\frac{4 a^{3} \cdot 7}{7} - 16 a - \frac{2 a (a + 2) (a - 2)}{7}$

Step 2.4

Write $- 16 a$ as a fraction with denominator $1$ .

$\frac{4 a^{3} \cdot 7}{7} + \frac{- 16 a}{1} - \frac{2 a (a + 2) (a - 2)}{7}$

Step 2.5

Multiply $\frac{- 16 a}{1}$ by $\frac{7}{7}$ .

$\frac{4 a^{3} \cdot 7}{7} + \frac{- 16 a}{1} \cdot \frac{7}{7} - \frac{2 a (a + 2) (a - 2)}{7}$

Step 2.6

Multiply $\frac{- 16 a}{1}$ by $\frac{7}{7}$ .

$\frac{4 a^{3} \cdot 7}{7} + \frac{- 16 a \cdot 7}{7} - \frac{2 a (a + 2) (a - 2)}{7}$

Step 3

Simplify terms.

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Step 3.1

Combine the numerators over the common denominator.

$\frac{4 a^{3} \cdot 7 - 16 a \cdot 7 - 2 a (a + 2) (a - 2)}{7}$

Step 3.2

Simplify each term.

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Step 3.2.1

Multiply $7$ by $4$ .

$\frac{28 a^{3} - 16 a \cdot 7 - 2 a (a + 2) (a - 2)}{7}$

Step 3.2.2

Multiply $7$ by $- 16$ .

$\frac{28 a^{3} - 112 a - 2 a (a + 2) (a - 2)}{7}$

Step 3.2.3

Apply the distributive property.

$\frac{28 a^{3} - 112 a + (- 2 a \cdot a - 2 a \cdot 2) (a - 2)}{7}$

Step 3.2.4

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

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Step 3.2.4.1

Move $a$ .

$\frac{28 a^{3} - 112 a + (- 2 (a \cdot a) - 2 a \cdot 2) (a - 2)}{7}$

Step 3.2.4.2

Multiply $a$ by $a$ .

$\frac{28 a^{3} - 112 a + (- 2 a^{2} - 2 a \cdot 2) (a - 2)}{7}$

Step 3.2.5

Multiply $2$ by $- 2$ .

$\frac{28 a^{3} - 112 a + (- 2 a^{2} - 4 a) (a - 2)}{7}$

Step 3.2.6

Expand $(- 2 a^{2} - 4 a) (a - 2)$ using the FOIL Method.

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Step 3.2.6.1

Apply the distributive property.

$\frac{28 a^{3} - 112 a - 2 a^{2} (a - 2) - 4 a (a - 2)}{7}$

Step 3.2.6.2

Apply the distributive property.

$\frac{28 a^{3} - 112 a - 2 a^{2} a - 2 a^{2} \cdot - 2 - 4 a (a - 2)}{7}$

Step 3.2.6.3

Apply the distributive property.

$\frac{28 a^{3} - 112 a - 2 a^{2} a - 2 a^{2} \cdot - 2 - 4 a \cdot a - 4 a \cdot - 2}{7}$

Step 3.2.7

Simplify and combine like terms.

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Step 3.2.7.1

Simplify each term.

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Step 3.2.7.1.1

Multiply $a^{2}$ by $a$ by adding the exponents.

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Step 3.2.7.1.1.1

Move $a$ .

$\frac{28 a^{3} - 112 a - 2 (a \cdot a^{2}) - 2 a^{2} \cdot - 2 - 4 a \cdot a - 4 a \cdot - 2}{7}$

Step 3.2.7.1.1.2

Multiply $a$ by $a^{2}$ .

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Step 3.2.7.1.1.2.1

Raise $a$ to the power of $1$ .

$\frac{28 a^{3} - 112 a - 2 (a^{1} a^{2}) - 2 a^{2} \cdot - 2 - 4 a \cdot a - 4 a \cdot - 2}{7}$

Step 3.2.7.1.1.2.2

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

$\frac{28 a^{3} - 112 a - 2 a^{1 + 2} - 2 a^{2} \cdot - 2 - 4 a \cdot a - 4 a \cdot - 2}{7}$

Step 3.2.7.1.1.3

Add $1$ and $2$ .

$\frac{28 a^{3} - 112 a - 2 a^{3} - 2 a^{2} \cdot - 2 - 4 a \cdot a - 4 a \cdot - 2}{7}$

Step 3.2.7.1.2

Multiply $- 2$ by $- 2$ .

$\frac{28 a^{3} - 112 a - 2 a^{3} + 4 a^{2} - 4 a \cdot a - 4 a \cdot - 2}{7}$

Step 3.2.7.1.3

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

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Step 3.2.7.1.3.1

Move $a$ .

$\frac{28 a^{3} - 112 a - 2 a^{3} + 4 a^{2} - 4 (a \cdot a) - 4 a \cdot - 2}{7}$

Step 3.2.7.1.3.2

Multiply $a$ by $a$ .

$\frac{28 a^{3} - 112 a - 2 a^{3} + 4 a^{2} - 4 a^{2} - 4 a \cdot - 2}{7}$

Step 3.2.7.1.4

Multiply $- 2$ by $- 4$ .

$\frac{28 a^{3} - 112 a - 2 a^{3} + 4 a^{2} - 4 a^{2} + 8 a}{7}$

Step 3.2.7.2

Subtract $4 a^{2}$ from $4 a^{2}$ .

$\frac{28 a^{3} - 112 a - 2 a^{3} + 0 + 8 a}{7}$

Step 3.2.7.3

Add $- 2 a^{3}$ and $0$ .

$\frac{28 a^{3} - 112 a - 2 a^{3} + 8 a}{7}$

Step 3.3

Simplify by adding terms.

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Step 3.3.1

Subtract $2 a^{3}$ from $28 a^{3}$ .

$\frac{26 a^{3} - 112 a + 8 a}{7}$

Step 3.3.2

Add $- 112 a$ and $8 a$ .

$\frac{26 a^{3} - 104 a}{7}$

Step 4

Simplify the numerator.

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Step 4.1

Factor $26 a$ out of $26 a^{3} - 104 a$ .

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Step 4.1.1

Factor $26 a$ out of $26 a^{3}$ .

$\frac{26 a (a^{2}) - 104 a}{7}$

Step 4.1.2

Factor $26 a$ out of $- 104 a$ .

$\frac{26 a (a^{2}) + 26 a (- 4)}{7}$

Step 4.1.3

Factor $26 a$ out of $26 a (a^{2}) + 26 a (- 4)$ .

$\frac{26 a (a^{2} - 4)}{7}$

Step 4.2

Rewrite $4$ as $2^{2}$ .

$\frac{26 a (a^{2} - 2^{2})}{7}$

Step 4.3

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

$\frac{26 a (a + 2) (a - 2)}{7}$