Simplify (2x+3)/(x^4)-(x+2)/(x^3)

$\frac{2 x + 3}{x^{4}} - \frac{x + 2}{x^{3}}$

Step 1

To write $- \frac{x + 2}{x^{3}}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{2 x + 3}{x^{4}} - \frac{x + 2}{x^{3}} \cdot \frac{x}{x}$

Step 2

Write each expression with a common denominator of $x^{4}$ , by multiplying each by an appropriate factor of $1$ .

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

$\frac{2 x + 3}{x^{4}} - \frac{(x + 2) x}{x^{3} x}$

Multiply $x^{3}$ by $x$ by adding the exponents.

Multiply $x^{3}$ by $x$ .

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

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

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

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

Add $3$ and $1$ .

$\frac{2 x + 3}{x^{4}} - \frac{(x + 2) x}{x^{4}}$

Step 3

Combine the numerators over the common denominator.

$\frac{2 x + 3 - (x + 2) x}{x^{4}}$

Step 4

Simplify the numerator.

Apply the distributive property.

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

Multiply $- 1$ by $2$ .

$\frac{2 x + 3 + (- x - 2) x}{x^{4}}$

Apply the distributive property.

$\frac{2 x + 3 - x \cdot x - 2 x}{x^{4}}$

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

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

Multiply $x$ by $x$ .

$\frac{2 x + 3 - x^{2} - 2 x}{x^{4}}$

Subtract $2 x$ from $2 x$ .

$\frac{3 - x^{2} + 0}{x^{4}}$

Add $3 - x^{2}$ and $0$ .

$\frac{3 - x^{2}}{x^{4}}$