Finite Math Examples

$\frac{r^{4} - 625}{r^{4}} \cdot \frac{r^{2} + a}{r^{2} + 25}$

Step 1

Simplify the numerator.

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Step 1.1

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

$\frac{{(r^{2})}^{2} - 625}{r^{4}} \cdot \frac{r^{2} + a}{r^{2} + 25}$

Step 1.2

Rewrite $625$ as $25^{2}$ .

$\frac{{(r^{2})}^{2} - 25^{2}}{r^{4}} \cdot \frac{r^{2} + a}{r^{2} + 25}$

Step 1.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = r^{2}$ and $b = 25$ .

$\frac{(r^{2} + 25) (r^{2} - 25)}{r^{4}} \cdot \frac{r^{2} + a}{r^{2} + 25}$

Step 1.4

Simplify.

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Step 1.4.1

Rewrite $25$ as $5^{2}$ .

$\frac{(r^{2} + 25) (r^{2} - 5^{2})}{r^{4}} \cdot \frac{r^{2} + a}{r^{2} + 25}$

Step 1.4.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = r$ and $b = 5$ .

$\frac{(r^{2} + 25) (r + 5) (r - 5)}{r^{4}} \cdot \frac{r^{2} + a}{r^{2} + 25}$

Step 2

Simplify terms.

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Step 2.1

Cancel the common factor of $r^{2} + 25$ .

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Step 2.1.1

Factor $r^{2} + 25$ out of $(r^{2} + 25) (r + 5) (r - 5)$ .

$\frac{(r^{2} + 25) ((r + 5) (r - 5))}{r^{4}} \cdot \frac{r^{2} + a}{r^{2} + 25}$

Step 2.1.2

Cancel the common factor.

$\frac{(r^{2} + 25) ((r + 5) (r - 5))}{r^{4}} \cdot \frac{r^{2} + a}{r^{2} + 25}$

Step 2.1.3

Rewrite the expression.

$\frac{(r + 5) (r - 5)}{r^{4}} \cdot (r^{2} + a)$

Step 2.2

Multiply $\frac{(r + 5) (r - 5)}{r^{4}}$ by $r^{2} + a$ .

$\frac{(r + 5) (r - 5) (r^{2} + a)}{r^{4}}$