Pre-Algebra Examples

$\frac{25 - \frac{1}{x^{2}}}{5 - \frac{1}{x}}$

Step 1

Multiply the numerator and denominator of the fraction by $x^{2}$ .

Tap for more steps...

Step 1.1

Multiply $\frac{25 - \frac{1}{x^{2}}}{5 - \frac{1}{x}}$ by $\frac{x^{2}}{x^{2}}$ .

$\frac{x^{2}}{x^{2}} \cdot \frac{25 - \frac{1}{x^{2}}}{5 - \frac{1}{x}}$

Step 1.2

Combine.

$\frac{x^{2} (25 - \frac{1}{x^{2}})}{x^{2} (5 - \frac{1}{x})}$

Step 2

Apply the distributive property.

$\frac{x^{2} \cdot 25 + x^{2} (- \frac{1}{x^{2}})}{x^{2} \cdot 5 + x^{2} (- \frac{1}{x})}$

Step 3

Simplify by cancelling.

Tap for more steps...

Step 3.1

Cancel the common factor of $x^{2}$ .

Tap for more steps...

Step 3.1.1

Move the leading negative in $- \frac{1}{x^{2}}$ into the numerator.

$\frac{x^{2} \cdot 25 + x^{2} \frac{- 1}{x^{2}}}{x^{2} \cdot 5 + x^{2} (- \frac{1}{x})}$

Step 3.1.2

Cancel the common factor.

$\frac{x^{2} \cdot 25 + x^{2} \frac{- 1}{x^{2}}}{x^{2} \cdot 5 + x^{2} (- \frac{1}{x})}$

Step 3.1.3

Rewrite the expression.

$\frac{x^{2} \cdot 25 - 1}{x^{2} \cdot 5 + x^{2} (- \frac{1}{x})}$

Step 3.2

Cancel the common factor of $x$ .

Tap for more steps...

Step 3.2.1

Move the leading negative in $- \frac{1}{x}$ into the numerator.

$\frac{x^{2} \cdot 25 - 1}{x^{2} \cdot 5 + x^{2} \frac{- 1}{x}}$

Step 3.2.2

Factor $x$ out of $x^{2}$ .

$\frac{x^{2} \cdot 25 - 1}{x^{2} \cdot 5 + x \cdot x \frac{- 1}{x}}$

Step 3.2.3

Cancel the common factor.

$\frac{x^{2} \cdot 25 - 1}{x^{2} \cdot 5 + x \cdot x \frac{- 1}{x}}$

Step 3.2.4

Rewrite the expression.

$\frac{x^{2} \cdot 25 - 1}{x^{2} \cdot 5 + x \cdot - 1}$

Step 4

Simplify the numerator.

Tap for more steps...

Step 4.1

Rewrite $x^{2} \cdot 25$ as ${(x \cdot 5)}^{2}$ .

$\frac{{(x \cdot 5)}^{2} - 1}{x^{2} \cdot 5 + x \cdot - 1}$

Step 4.2

Rewrite $1$ as $1^{2}$ .

$\frac{{(x \cdot 5)}^{2} - 1^{2}}{x^{2} \cdot 5 + x \cdot - 1}$

Step 4.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x \cdot 5$ and $b = 1$ .

$\frac{(x \cdot 5 + 1) (x \cdot 5 - 1)}{x^{2} \cdot 5 + x \cdot - 1}$

Step 4.4

Simplify.

Tap for more steps...

Step 4.4.1

Move $5$ to the left of $x$ .

$\frac{(5 \cdot x + 1) (x \cdot 5 - 1)}{x^{2} \cdot 5 + x \cdot - 1}$

Step 4.4.2

Move $5$ to the left of $x$ .

$\frac{(5 x + 1) (5 x - 1)}{x^{2} \cdot 5 + x \cdot - 1}$

Step 5

Simplify the denominator.

Tap for more steps...

Step 5.1

Factor $x$ out of $x^{2} \cdot 5 + x \cdot - 1$ .

Tap for more steps...

Step 5.1.1

Factor $x$ out of $x^{2} \cdot 5$ .

$\frac{(5 x + 1) (5 x - 1)}{x (x \cdot 5) + x \cdot - 1}$

Step 5.1.2

Factor $x$ out of $x \cdot - 1$ .

$\frac{(5 x + 1) (5 x - 1)}{x (x \cdot 5) + x (- 1)}$

Step 5.1.3

Factor $x$ out of $x (x \cdot 5) + x (- 1)$ .

$\frac{(5 x + 1) (5 x - 1)}{x (x \cdot 5 - 1)}$

Step 5.2

Move $5$ to the left of $x$ .

$\frac{(5 x + 1) (5 x - 1)}{x (5 x - 1)}$

Step 6

Cancel the common factor of $5 x - 1$ .

Tap for more steps...

Step 6.1

Cancel the common factor.

$\frac{(5 x + 1) (5 x - 1)}{x (5 x - 1)}$

Step 6.2

Rewrite the expression.

$\frac{5 x + 1}{x}$

Step 7

Split the fraction $\frac{5 x + 1}{x}$ into two fractions.

$\frac{5 x}{x} + \frac{1}{x}$

Step 8

Cancel the common factor of $x$ .

Tap for more steps...

Step 8.1

Cancel the common factor.

$\frac{5 x}{x} + \frac{1}{x}$

Step 8.2

Divide $5$ by $1$ .

$5 + \frac{1}{x}$