Simplify (2x+1)/(8x^3)+(x-1)/(4x^2)

$\frac{2 x + 1}{8 x^{3}} + \frac{x - 1}{4 x^{2}}$

Step 1

To write $\frac{x - 1}{4 x^{2}}$ as a fraction with a common denominator, multiply by $\frac{2 x}{2 x}$ .

$\frac{2 x + 1}{8 x^{3}} + \frac{x - 1}{4 x^{2}} \cdot \frac{2 x}{2 x}$

Step 2

Write each expression with a common denominator of $8 x^{3}$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{x - 1}{4 x^{2}}$ by $\frac{2 x}{2 x}$ .

$\frac{2 x + 1}{8 x^{3}} + \frac{(x - 1) (2 x)}{4 x^{2} (2 x)}$

Multiply $2$ by $4$ .

$\frac{2 x + 1}{8 x^{3}} + \frac{(x - 1) (2 x)}{8 x^{2} x}$

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

$\frac{2 x + 1}{8 x^{3}} + \frac{(x - 1) (2 x)}{8 (x^{1} x^{2})}$

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

$\frac{2 x + 1}{8 x^{3}} + \frac{(x - 1) (2 x)}{8 x^{1 + 2}}$

Add $1$ and $2$ .

$\frac{2 x + 1}{8 x^{3}} + \frac{(x - 1) (2 x)}{8 x^{3}}$

Step 3

Combine the numerators over the common denominator.

$\frac{2 x + 1 + (x - 1) (2 x)}{8 x^{3}}$

Step 4

Simplify the numerator.

Apply the distributive property.

$\frac{2 x + 1 + (x \cdot 2 - 1 \cdot 2) x}{8 x^{3}}$

Move $2$ to the left of $x$ .

$\frac{2 x + 1 + (2 \cdot x - 1 \cdot 2) x}{8 x^{3}}$

Multiply $- 1$ by $2$ .

$\frac{2 x + 1 + (2 \cdot x - 2) x}{8 x^{3}}$

Apply the distributive property.

$\frac{2 x + 1 + 2 x \cdot x - 2 x}{8 x^{3}}$

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

Move $x$ .

$\frac{2 x + 1 + 2 (x \cdot x) - 2 x}{8 x^{3}}$

Multiply $x$ by $x$ .

$\frac{2 x + 1 + 2 x^{2} - 2 x}{8 x^{3}}$

Subtract $2 x$ from $2 x$ .

$\frac{1 + 2 x^{2} + 0}{8 x^{3}}$

Add $1 + 2 x^{2}$ and $0$ .

$\frac{1 + 2 x^{2}}{8 x^{3}}$