Basic Math Examples

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

Step 1

Multiply the numerator by the reciprocal of the denominator.

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

Step 2

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

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

Step 3

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 $a = a$ and $b = b$ .

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

Step 4

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 $b = b$ .

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

Step 5

Combine.

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

Step 6

Multiply $a + b$ by ${(a + b)}^{2}$ by adding the exponents.

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Step 6.1

Move ${(a + b)}^{2}$ .

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

Step 6.2

Multiply ${(a + b)}^{2}$ by $a + b$ .

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Step 6.2.1

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

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

Step 6.2.2

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

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

Step 6.3

Add $2$ and $1$ .

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

Step 7

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

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Step 7.1

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

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

Step 7.2

Cancel the common factors.

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Step 7.2.1

Factor $a + b$ out of ${(a + b)}^{3} (a^{2} - a b + b^{2})$ .

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

Step 7.2.2

Cancel the common factor.

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

Step 7.2.3

Rewrite the expression.

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

Step 8

Simplify the numerator.

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Step 8.1

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

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

Step 8.2

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

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

Step 8.3

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

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

Step 8.4

Add $1$ and $1$ .

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

Step 9

Simplify terms.

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Step 9.1

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

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Step 9.1.1

Factor ${(a - b)}^{2}$ out of ${(a - b)}^{3}$ .

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

Step 9.1.2

Cancel the common factor.

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

Step 9.1.3

Rewrite the expression.

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

Step 9.2

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

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

Step 10

Simplify the numerator.

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Step 10.1

Rewrite $a^{4}$ as ${(a^{2})}^{2}$ .

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

Step 10.2

Rewrite $b^{4}$ as ${(b^{2})}^{2}$ .

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

Step 10.3

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

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

Step 10.4

Factor.

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

Step 11

Reduce the expression by cancelling the common factors.

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Step 11.1

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

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Step 11.1.1

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

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

Step 11.1.2

Cancel the common factors.

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Step 11.1.2.1

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

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

Step 11.1.2.2

Cancel the common factor.

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

Step 11.1.2.3

Rewrite the expression.

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

Step 11.2

Cancel the common factor of $a - b$ .

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Step 11.2.1

Cancel the common factor.

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

Step 11.2.2

Rewrite the expression.

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