Simplify 6/(3x)+(x+5)/(x^3)+(1-2x^2)/(x^4)

$\frac{6}{3 x} + \frac{x + 5}{x^{3}} + \frac{1 - 2 x^{2}}{x^{4}}$

Step 1

Find the common denominator.

Multiply $\frac{6}{3 x}$ by $\frac{x^{3}}{x^{3}}$ .

$\frac{6}{3 x} \cdot \frac{x^{3}}{x^{3}} + \frac{x + 5}{x^{3}} + \frac{1 - 2 x^{2}}{x^{4}}$

Multiply $\frac{6}{3 x}$ by $\frac{x^{3}}{x^{3}}$ .

$\frac{6 x^{3}}{3 x \cdot x^{3}} + \frac{x + 5}{x^{3}} + \frac{1 - 2 x^{2}}{x^{4}}$

Multiply $\frac{x + 5}{x^{3}}$ by $\frac{3 x}{3 x}$ .

$\frac{6 x^{3}}{3 x \cdot x^{3}} + \frac{x + 5}{x^{3}} \cdot \frac{3 x}{3 x} + \frac{1 - 2 x^{2}}{x^{4}}$

Multiply $\frac{x + 5}{x^{3}}$ by $\frac{3 x}{3 x}$ .

$\frac{6 x^{3}}{3 x \cdot x^{3}} + \frac{(x + 5) (3 x)}{x^{3} (3 x)} + \frac{1 - 2 x^{2}}{x^{4}}$

Multiply $\frac{1 - 2 x^{2}}{x^{4}}$ by $\frac{3}{3}$ .

$\frac{6 x^{3}}{3 x \cdot x^{3}} + \frac{(x + 5) (3 x)}{x^{3} (3 x)} + \frac{1 - 2 x^{2}}{x^{4}} \cdot \frac{3}{3}$

Multiply $\frac{1 - 2 x^{2}}{x^{4}}$ by $\frac{3}{3}$ .

$\frac{6 x^{3}}{3 x \cdot x^{3}} + \frac{(x + 5) (3 x)}{x^{3} (3 x)} + \frac{(1 - 2 x^{2}) \cdot 3}{x^{4} \cdot 3}$

Reorder the factors of $3 x \cdot x^{3}$ .

$\frac{6 x^{3}}{3 x^{3} x} + \frac{(x + 5) (3 x)}{x^{3} (3 x)} + \frac{(1 - 2 x^{2}) \cdot 3}{x^{4} \cdot 3}$

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

$\frac{6 x^{3}}{3 (x^{1} x^{3})} + \frac{(x + 5) (3 x)}{x^{3} (3 x)} + \frac{(1 - 2 x^{2}) \cdot 3}{x^{4} \cdot 3}$

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

$\frac{6 x^{3}}{3 x^{1 + 3}} + \frac{(x + 5) (3 x)}{x^{3} (3 x)} + \frac{(1 - 2 x^{2}) \cdot 3}{x^{4} \cdot 3}$

Add $1$ and $3$ .

$\frac{6 x^{3}}{3 x^{4}} + \frac{(x + 5) (3 x)}{x^{3} (3 x)} + \frac{(1 - 2 x^{2}) \cdot 3}{x^{4} \cdot 3}$

Reorder the factors of $x^{3} (3 x)$ .

$\frac{6 x^{3}}{3 x^{4}} + \frac{(x + 5) (3 x)}{3 x^{3} x} + \frac{(1 - 2 x^{2}) \cdot 3}{x^{4} \cdot 3}$

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

$\frac{6 x^{3}}{3 x^{4}} + \frac{(x + 5) (3 x)}{3 (x^{1} x^{3})} + \frac{(1 - 2 x^{2}) \cdot 3}{x^{4} \cdot 3}$

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

$\frac{6 x^{3}}{3 x^{4}} + \frac{(x + 5) (3 x)}{3 x^{1 + 3}} + \frac{(1 - 2 x^{2}) \cdot 3}{x^{4} \cdot 3}$

Add $1$ and $3$ .

$\frac{6 x^{3}}{3 x^{4}} + \frac{(x + 5) (3 x)}{3 x^{4}} + \frac{(1 - 2 x^{2}) \cdot 3}{x^{4} \cdot 3}$

Reorder the factors of $x^{4} \cdot 3$ .

$\frac{6 x^{3}}{3 x^{4}} + \frac{(x + 5) (3 x)}{3 x^{4}} + \frac{(1 - 2 x^{2}) \cdot 3}{3 x^{4}}$

Step 2

Combine the numerators over the common denominator.

$\frac{6 x^{3} + (x + 5) (3 x) + (1 - 2 x^{2}) \cdot 3}{3 x^{4}}$

Step 3

Simplify each term.

Apply the distributive property.

$\frac{6 x^{3} + x (3 x) + 5 (3 x) + (1 - 2 x^{2}) \cdot 3}{3 x^{4}}$

Rewrite using the commutative property of multiplication.

$\frac{6 x^{3} + 3 x \cdot x + 5 (3 x) + (1 - 2 x^{2}) \cdot 3}{3 x^{4}}$

Multiply $3$ by $5$ .

$\frac{6 x^{3} + 3 x \cdot x + 15 x + (1 - 2 x^{2}) \cdot 3}{3 x^{4}}$

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

Move $x$ .

$\frac{6 x^{3} + 3 (x \cdot x) + 15 x + (1 - 2 x^{2}) \cdot 3}{3 x^{4}}$

Multiply $x$ by $x$ .

$\frac{6 x^{3} + 3 x^{2} + 15 x + (1 - 2 x^{2}) \cdot 3}{3 x^{4}}$

Apply the distributive property.

$\frac{6 x^{3} + 3 x^{2} + 15 x + 1 \cdot 3 - 2 x^{2} \cdot 3}{3 x^{4}}$

Multiply $3$ by $1$ .

$\frac{6 x^{3} + 3 x^{2} + 15 x + 3 - 2 x^{2} \cdot 3}{3 x^{4}}$

Multiply $3$ by $- 2$ .

$\frac{6 x^{3} + 3 x^{2} + 15 x + 3 - 6 x^{2}}{3 x^{4}}$

Step 4

Simplify terms.

Subtract $6 x^{2}$ from $3 x^{2}$ .

$\frac{6 x^{3} - 3 x^{2} + 15 x + 3}{3 x^{4}}$

Cancel the common factor of $6 x^{3} - 3 x^{2} + 15 x + 3$ and $3$ .

Factor $3$ out of $6 x^{3}$ .

$\frac{3 (2 x^{3}) - 3 x^{2} + 15 x + 3}{3 x^{4}}$

Factor $3$ out of $- 3 x^{2}$ .

$\frac{3 (2 x^{3}) + 3 (- x^{2}) + 15 x + 3}{3 x^{4}}$

Factor $3$ out of $3 (2 x^{3}) + 3 (- x^{2})$ .

$\frac{3 (2 x^{3} - x^{2}) + 15 x + 3}{3 x^{4}}$

Factor $3$ out of $15 x$ .

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

Factor $3$ out of $3 (2 x^{3} - x^{2}) + 3 (5 x)$ .

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

Factor $3$ out of $3$ .

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

Factor $3$ out of $3 (2 x^{3} - x^{2} + 5 x) + 3 (1)$ .

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Simplify 6/(3x)+(x+5)/(x^3)+(1-2x^2)/(x^4)

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