Finite Math Examples

$\frac{8 s}{9 s^{2} - 25 t^{2}} - \frac{s}{3 s - 5 t}$

Step 1

Simplify the denominator.

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Step 1.1

Rewrite $9 s^{2}$ as ${(3 s)}^{2}$ .

$\frac{8 s}{{(3 s)}^{2} - 25 t^{2}} - \frac{s}{3 s - 5 t}$

Step 1.2

Rewrite $25 t^{2}$ as ${(5 t)}^{2}$ .

$\frac{8 s}{{(3 s)}^{2} - {(5 t)}^{2}} - \frac{s}{3 s - 5 t}$

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 $a = 3 s$ and $b = 5 t$ .

$\frac{8 s}{(3 s + 5 t) (3 s - (5 t))} - \frac{s}{3 s - 5 t}$

Step 1.4

Multiply $5$ by $- 1$ .

$\frac{8 s}{(3 s + 5 t) (3 s - 5 t)} - \frac{s}{3 s - 5 t}$

Step 2

To write $- \frac{s}{3 s - 5 t}$ as a fraction with a common denominator, multiply by $\frac{3 s + 5 t}{3 s + 5 t}$ .

$\frac{8 s}{(3 s + 5 t) (3 s - 5 t)} - \frac{s}{3 s - 5 t} \cdot \frac{3 s + 5 t}{3 s + 5 t}$

Step 3

Write each expression with a common denominator of $(3 s + 5 t) (3 s - 5 t)$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{s}{3 s - 5 t}$ by $\frac{3 s + 5 t}{3 s + 5 t}$ .

$\frac{8 s}{(3 s + 5 t) (3 s - 5 t)} - \frac{s (3 s + 5 t)}{(3 s - 5 t) (3 s + 5 t)}$

Step 3.2

Reorder the factors of $(3 s - 5 t) (3 s + 5 t)$ .

$\frac{8 s}{(3 s + 5 t) (3 s - 5 t)} - \frac{s (3 s + 5 t)}{(3 s + 5 t) (3 s - 5 t)}$

Step 4

Combine the numerators over the common denominator.

$\frac{8 s - s (3 s + 5 t)}{(3 s + 5 t) (3 s - 5 t)}$

Step 5

Simplify the numerator.

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Step 5.1

Factor $s$ out of $8 s - s (3 s + 5 t)$ .

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Step 5.1.1

Factor $s$ out of $8 s$ .

$\frac{s \cdot 8 - s (3 s + 5 t)}{(3 s + 5 t) (3 s - 5 t)}$

Step 5.1.2

Factor $s$ out of $- s (3 s + 5 t)$ .

$\frac{s \cdot 8 + s (- (3 s + 5 t))}{(3 s + 5 t) (3 s - 5 t)}$

Step 5.1.3

Factor $s$ out of $s \cdot 8 + s (- (3 s + 5 t))$ .

$\frac{s (8 - (3 s + 5 t))}{(3 s + 5 t) (3 s - 5 t)}$

Step 5.2

Apply the distributive property.

$\frac{s (8 - (3 s) - (5 t))}{(3 s + 5 t) (3 s - 5 t)}$

Step 5.3

Multiply $3$ by $- 1$ .

$\frac{s (8 - 3 s - (5 t))}{(3 s + 5 t) (3 s - 5 t)}$

Step 5.4

Multiply $5$ by $- 1$ .

$\frac{s (8 - 3 s - 5 t)}{(3 s + 5 t) (3 s - 5 t)}$