Simplify (1/x+3/(x^2))/(x+27/(x^2))

$\frac{\frac{1}{x} + \frac{3}{x^{2}}}{x + \frac{27}{x^{2}}}$

Step 1

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

Multiply $\frac{\frac{1}{x} + \frac{3}{x^{2}}}{x + \frac{27}{x^{2}}}$ by $\frac{x^{2}}{x^{2}}$ .

$\frac{x^{2}}{x^{2}} \cdot \frac{\frac{1}{x} + \frac{3}{x^{2}}}{x + \frac{27}{x^{2}}}$

Combine.

$\frac{x^{2} (\frac{1}{x} + \frac{3}{x^{2}})}{x^{2} (x + \frac{27}{x^{2}})}$

Step 2

Apply the distributive property.

$\frac{x^{2} \frac{1}{x} + x^{2} \frac{3}{x^{2}}}{x^{2} x + x^{2} \frac{27}{x^{2}}}$

Step 3

Simplify by cancelling.

Cancel the common factor of $x$ .

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

$\frac{x \cdot x \frac{1}{x} + x^{2} \frac{3}{x^{2}}}{x^{2} x + x^{2} \frac{27}{x^{2}}}$

Cancel the common factor.

$\frac{x \cdot x \frac{1}{x} + x^{2} \frac{3}{x^{2}}}{x^{2} x + x^{2} \frac{27}{x^{2}}}$

Rewrite the expression.

$\frac{x + x^{2} \frac{3}{x^{2}}}{x^{2} x + x^{2} \frac{27}{x^{2}}}$

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

Cancel the common factor.

$\frac{x + x^{2} \frac{3}{x^{2}}}{x^{2} x + x^{2} \frac{27}{x^{2}}}$

Rewrite the expression.

$\frac{x + 3}{x^{2} x + x^{2} \frac{27}{x^{2}}}$

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

Cancel the common factor.

$\frac{x + 3}{x^{2} x + x^{2} \frac{27}{x^{2}}}$

Rewrite the expression.

$\frac{x + 3}{x^{2} x + 27}$

Step 4

Simplify the denominator.

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

Multiply $x^{2}$ by $x$ .

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

$\frac{x + 3}{x^{2} x^{1} + 27}$

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

$\frac{x + 3}{x^{2 + 1} + 27}$

Add $2$ and $1$ .

$\frac{x + 3}{x^{3} + 27}$

Rewrite $27$ as $3^{3}$ .

$\frac{x + 3}{x^{3} + 3^{3}}$

Since both terms are perfect cubes, factor using the sum of cubes formula, $a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$ where $a = x$ and $b = 3$ .

$\frac{x + 3}{(x + 3) (x^{2} - x \cdot 3 + 3^{2})}$

Simplify.

Multiply $3$ by $- 1$ .

$\frac{x + 3}{(x + 3) (x^{2} - 3 x + 3^{2})}$

Raise $3$ to the power of $2$ .

$\frac{x + 3}{(x + 3) (x^{2} - 3 x + 9)}$

Step 5

Cancel the common factor of $x + 3$ .

Cancel the common factor.

$\frac{x + 3}{(x + 3) (x^{2} - 3 x + 9)}$

Rewrite the expression.

$\frac{1}{x^{2} - 3 x + 9}$

Simplify (1/x+3/(x^2))/(x+27/(x^2))

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