Algebra Examples

$\frac{a^{2} - 4}{3 a} \div \frac{5 a^{2}}{a - 2}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{a^{2} - 4}{3 a} \cdot \frac{a - 2}{5 a^{2}}$

Step 2

Simplify the numerator.

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Step 2.1

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

$\frac{a^{2} - 2^{2}}{3 a} \cdot \frac{a - 2}{5 a^{2}}$

Step 2.2

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 $b = 2$ .

$\frac{(a + 2) (a - 2)}{3 a} \cdot \frac{a - 2}{5 a^{2}}$

Step 3

Multiply $\frac{(a + 2) (a - 2)}{3 a} \cdot \frac{a - 2}{5 a^{2}}$ .

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Step 3.1

Multiply $\frac{(a + 2) (a - 2)}{3 a}$ by $\frac{a - 2}{5 a^{2}}$ .

$\frac{(a + 2) (a - 2) (a - 2)}{3 a (5 a^{2})}$

Step 3.2

Raise $a - 2$ to the power of $1$ .

$\frac{(a + 2) ({(a - 2)}^{1} (a - 2))}{3 a (5 a^{2})}$

Step 3.3

Raise $a - 2$ to the power of $1$ .

$\frac{(a + 2) ({(a - 2)}^{1} {(a - 2)}^{1})}{3 a (5 a^{2})}$

Step 3.4

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

$\frac{(a + 2) {(a - 2)}^{1 + 1}}{3 a (5 a^{2})}$

Step 3.5

Add $1$ and $1$ .

$\frac{(a + 2) {(a - 2)}^{2}}{3 a (5 a^{2})}$

Step 3.6

Multiply $5$ by $3$ .

$\frac{(a + 2) {(a - 2)}^{2}}{15 a \cdot a^{2}}$

Step 3.7

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

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Step 3.7.1

Move $a^{2}$ .

$\frac{(a + 2) {(a - 2)}^{2}}{15 (a^{2} a)}$

Step 3.7.2

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

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Step 3.7.2.1

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

$\frac{(a + 2) {(a - 2)}^{2}}{15 (a^{2} a^{1})}$

Step 3.7.2.2

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

$\frac{(a + 2) {(a - 2)}^{2}}{15 a^{2 + 1}}$

Step 3.7.3

Add $2$ and $1$ .

$\frac{(a + 2) {(a - 2)}^{2}}{15 a^{3}}$