Simplify (4/x+7)/(16/(x^2)-49)

$\frac{\frac{4}{x} + 7}{\frac{16}{x^{2}} - 49}$

Step 1

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

Multiply $\frac{\frac{4}{x} + 7}{\frac{16}{x^{2}} - 49}$ by $\frac{x^{2}}{x^{2}}$ .

$\frac{x^{2}}{x^{2}} \cdot \frac{\frac{4}{x} + 7}{\frac{16}{x^{2}} - 49}$

Combine.

$\frac{x^{2} (\frac{4}{x} + 7)}{x^{2} (\frac{16}{x^{2}} - 49)}$

Step 2

Apply the distributive property.

$\frac{x^{2} \frac{4}{x} + x^{2} \cdot 7}{x^{2} \frac{16}{x^{2}} + x^{2} \cdot - 49}$

Step 3

Simplify by cancelling.

Cancel the common factor of $x$ .

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

$\frac{x \cdot x \frac{4}{x} + x^{2} \cdot 7}{x^{2} \frac{16}{x^{2}} + x^{2} \cdot - 49}$

Cancel the common factor.

$\frac{x \cdot x \frac{4}{x} + x^{2} \cdot 7}{x^{2} \frac{16}{x^{2}} + x^{2} \cdot - 49}$

Rewrite the expression.

$\frac{x \cdot 4 + x^{2} \cdot 7}{x^{2} \frac{16}{x^{2}} + x^{2} \cdot - 49}$

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

Cancel the common factor.

$\frac{x \cdot 4 + x^{2} \cdot 7}{x^{2} \frac{16}{x^{2}} + x^{2} \cdot - 49}$

Rewrite the expression.

$\frac{x \cdot 4 + x^{2} \cdot 7}{16 + x^{2} \cdot - 49}$

Step 4

Simplify the numerator.

Factor $x$ out of $x \cdot 4 + x^{2} \cdot 7$ .

Factor $x$ out of $x \cdot 4$ .

$\frac{x (4) + x^{2} \cdot 7}{16 + x^{2} \cdot - 49}$

Factor $x$ out of $x^{2} \cdot 7$ .

$\frac{x (4) + x (x \cdot 7)}{16 + x^{2} \cdot - 49}$

Factor $x$ out of $x (4) + x (x \cdot 7)$ .

$\frac{x (4 + x \cdot 7)}{16 + x^{2} \cdot - 49}$

Move $7$ to the left of $x$ .

$\frac{x (4 + 7 x)}{16 + x^{2} \cdot - 49}$

Step 5

Simplify the denominator.

Rewrite $16$ as $4^{2}$ .

$\frac{x (4 + 7 x)}{4^{2} + x^{2} \cdot - 49}$

Rewrite $- x^{2} \cdot - 49$ as ${(7 x)}^{2}$ .

$\frac{x (4 + 7 x)}{4^{2} - {(7 x)}^{2}}$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 4$ and $b = 7 x$ .

$\frac{x (4 + 7 x)}{(4 + 7 x) (4 - (7 x))}$

Multiply $7$ by $- 1$ .

$\frac{x (4 + 7 x)}{(4 + 7 x) (4 - 7 x)}$

Step 6

Cancel the common factor of $4 + 7 x$ .

Cancel the common factor.

$\frac{x (4 + 7 x)}{(4 + 7 x) (4 - 7 x)}$

Rewrite the expression.

$\frac{x}{4 - 7 x}$

Simplify (4/x+7)/(16/(x^2)-49)

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