Simplify (1-25/(x^2))/((x^2)/25+x/5)

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

Multiply the numerator and denominator of the complex fraction by $25 x^{2}$ .

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

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

Combine.

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

Apply the distributive property.

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

Simplify by cancelling.

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

Move the leading negative in $- \frac{25}{x^{2}}$ into the numerator.

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

Factor $x^{2}$ out of $25 x^{2}$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply $25$ by $- 25$ .

$\frac{25 x^{2} \cdot 1 - 625}{25 x^{2} \frac{x^{2}}{25} + 25 x^{2} \frac{x}{5}}$

Cancel the common factor of $25$ .

Factor $25$ out of $25 x^{2}$ .

$\frac{25 x^{2} \cdot 1 - 625}{25 (x^{2}) \frac{x^{2}}{25} + 25 x^{2} \frac{x}{5}}$

Cancel the common factor.

$\frac{25 x^{2} \cdot 1 - 625}{25 x^{2} \frac{x^{2}}{25} + 25 x^{2} \frac{x}{5}}$

Rewrite the expression.

$\frac{25 x^{2} \cdot 1 - 625}{x^{2} x^{2} + 25 x^{2} \frac{x}{5}}$

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

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

$\frac{25 x^{2} \cdot 1 - 625}{x^{2 + 2} + 25 x^{2} \frac{x}{5}}$

Add $2$ and $2$ .

$\frac{25 x^{2} \cdot 1 - 625}{x^{4} + 25 x^{2} \frac{x}{5}}$

Cancel the common factor of $5$ .

Factor $5$ out of $25 x^{2}$ .

$\frac{25 x^{2} \cdot 1 - 625}{x^{4} + 5 (5 x^{2}) \frac{x}{5}}$

Cancel the common factor.

$\frac{25 x^{2} \cdot 1 - 625}{x^{4} + 5 (5 x^{2}) \frac{x}{5}}$

Rewrite the expression.

$\frac{25 x^{2} \cdot 1 - 625}{x^{4} + 5 x^{2} x}$

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

$\frac{25 x^{2} \cdot 1 - 625}{x^{4} + 5 (x^{1} x^{2})}$

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

$\frac{25 x^{2} \cdot 1 - 625}{x^{4} + 5 x^{1 + 2}}$

Add $1$ and $2$ .

$\frac{25 x^{2} \cdot 1 - 625}{x^{4} + 5 x^{3}}$

Simplify the numerator.

Factor $25$ out of $25 x^{2} \cdot 1 - 625$ .

Factor $25$ out of $25 x^{2} \cdot 1$ .

$\frac{25 (x^{2} \cdot 1) - 625}{x^{4} + 5 x^{3}}$

Factor $25$ out of $- 625$ .

$\frac{25 (x^{2} \cdot 1) + 25 \cdot - 25}{x^{4} + 5 x^{3}}$

Factor $25$ out of $25 (x^{2} \cdot 1) + 25 \cdot - 25$ .

$\frac{25 (x^{2} \cdot 1 - 25)}{x^{4} + 5 x^{3}}$

Rewrite $x^{2} \cdot 1$ as ${(x \cdot 1)}^{2}$ .

$\frac{25 ({(x \cdot 1)}^{2} - 25)}{x^{4} + 5 x^{3}}$

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

$\frac{25 ({(x \cdot 1)}^{2} - 5^{2})}{x^{4} + 5 x^{3}}$

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

$\frac{25 ((x \cdot 1 + 5) (x \cdot 1 - 5))}{x^{4} + 5 x^{3}}$

Simplify.

Multiply $x$ by $1$ .

$\frac{25 ((x + 5) (x \cdot 1 - 5))}{x^{4} + 5 x^{3}}$

Multiply $x$ by $1$ .

$\frac{25 ((x + 5) (x - 5))}{x^{4} + 5 x^{3}}$

$\frac{25 (x + 5) (x - 5)}{x^{4} + 5 x^{3}}$

Simplify terms.

Factor $x^{3}$ out of $x^{4} + 5 x^{3}$ .

Factor $x^{3}$ out of $x^{4}$ .

$\frac{25 (x + 5) (x - 5)}{x^{3} x + 5 x^{3}}$

Factor $x^{3}$ out of $5 x^{3}$ .

$\frac{25 (x + 5) (x - 5)}{x^{3} x + x^{3} \cdot 5}$

Factor $x^{3}$ out of $x^{3} x + x^{3} \cdot 5$ .

$\frac{25 (x + 5) (x - 5)}{x^{3} (x + 5)}$

Cancel the common factor of $x + 5$ .

Cancel the common factor.

$\frac{25 (x + 5) (x - 5)}{x^{3} (x + 5)}$

Rewrite the expression.

$\frac{25 (x - 5)}{x^{3}}$

Simplify (1-25/(x^2))/((x^2)/25+x/5)

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