Basic Math Examples

$\frac{\frac{a^{7}}{8 b^{3} \cdot (a^{2} - b^{2})}}{\frac{(a - b) \cdot a^{- 1}}{8 b^{2} \cdot (a + b)}}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

$\frac{a^{7}}{8 b^{3} \cdot (a^{2} - b^{2})} \cdot \frac{8 b^{2} \cdot (a + b)}{(a - b) \cdot a^{- 1}}$

Step 2

Combine.

$\frac{a^{7} (8 b^{2} \cdot (a + b))}{8 b^{3} \cdot (a^{2} - b^{2}) ((a - b) \cdot a^{- 1})}$

Step 3

Move $a^{- 1}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$\frac{a^{7} (8 b^{2} \cdot (a + b)) a}{8 b^{3} \cdot (a^{2} - b^{2}) (a - b)}$

Step 4

Multiply $a^{7}$ by $a$ by adding the exponents.

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Step 4.1

Move $a$ .

$\frac{a \cdot a^{7} (8 b^{2} \cdot (a + b))}{8 b^{3} \cdot (a^{2} - b^{2}) (a - b)}$

Step 4.2

Multiply $a$ by $a^{7}$ .

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Step 4.2.1

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

$\frac{a^{1} a^{7} (8 b^{2} \cdot (a + b))}{8 b^{3} \cdot (a^{2} - b^{2}) (a - b)}$

Step 4.2.2

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

$\frac{a^{1 + 7} (8 b^{2} \cdot (a + b))}{8 b^{3} \cdot (a^{2} - b^{2}) (a - b)}$

Step 4.3

Add $1$ and $7$ .

$\frac{a^{8} (8 b^{2} \cdot (a + b))}{8 b^{3} \cdot (a^{2} - b^{2}) (a - b)}$

Step 5

Reduce the expression by cancelling the common factors.

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Step 5.1

Cancel the common factor of $8$ .

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Step 5.1.1

Cancel the common factor.

$\frac{a^{8} (8 b^{2} \cdot (a + b))}{8 b^{3} \cdot (a^{2} - b^{2}) (a - b)}$

Step 5.1.2

Rewrite the expression.

$\frac{a^{8} (b^{2} \cdot (a + b))}{(b^{3} \cdot (a^{2} - b^{2})) (a - b)}$

Step 5.2

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

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Step 5.2.1

Factor $b^{2}$ out of $a^{8} (b^{2} \cdot (a + b))$ .

$\frac{b^{2} (a^{8} (a + b))}{(b^{3} \cdot (a^{2} - b^{2})) (a - b)}$

Step 5.2.2

Cancel the common factors.

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Step 5.2.2.1

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

$\frac{b^{2} (a^{8} (a + b))}{b^{2} ((b \cdot (a^{2} - b^{2})) (a - b))}$

Step 5.2.2.2

Cancel the common factor.

$\frac{b^{2} (a^{8} (a + b))}{b^{2} ((b \cdot (a^{2} - b^{2})) (a - b))}$

Step 5.2.2.3

Rewrite the expression.

$\frac{a^{8} (a + b)}{(b \cdot (a^{2} - b^{2})) (a - b)}$

Step 6

Simplify the denominator.

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Step 6.1

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 $b = b$ .

$\frac{a^{8} (a + b)}{b \cdot (a + b) (a - b) (a - b)}$

Step 6.2

Combine exponents.

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Step 6.2.1

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

$\frac{a^{8} (a + b)}{b \cdot (a + b) ({(a - b)}^{1} (a - b))}$

Step 6.2.2

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

$\frac{a^{8} (a + b)}{b \cdot (a + b) ({(a - b)}^{1} {(a - b)}^{1})}$

Step 6.2.3

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

$\frac{a^{8} (a + b)}{b \cdot (a + b) {(a - b)}^{1 + 1}}$

Step 6.2.4

Add $1$ and $1$ .

$\frac{a^{8} (a + b)}{b \cdot (a + b) {(a - b)}^{2}}$

Step 7

Cancel the common factor of $a + b$ .

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Step 7.1

Cancel the common factor.

$\frac{a^{8} (a + b)}{b \cdot (a + b) {(a - b)}^{2}}$

Step 7.2

Rewrite the expression.

$\frac{a^{8}}{b {(a - b)}^{2}}$