Algebra Examples

$\frac{4}{3 x} + \frac{8}{x^{3}} + \frac{2}{5 x y}$

Step 1

To write $\frac{4}{3 x}$ as a fraction with a common denominator, multiply by $\frac{x^{2}}{x^{2}}$ .

$\frac{4}{3 x} \cdot \frac{x^{2}}{x^{2}} + \frac{8}{x^{3}} + \frac{2}{5 x y}$

Step 2

To write $\frac{8}{x^{3}}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{4}{3 x} \cdot \frac{x^{2}}{x^{2}} + \frac{8}{x^{3}} \cdot \frac{3}{3} + \frac{2}{5 x y}$

Step 3

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

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Step 3.1

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

$\frac{4 x^{2}}{3 x \cdot x^{2}} + \frac{8}{x^{3}} \cdot \frac{3}{3} + \frac{2}{5 x y}$

Step 3.2

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

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Step 3.2.1

Move $x^{2}$ .

$\frac{4 x^{2}}{3 (x^{2} x)} + \frac{8}{x^{3}} \cdot \frac{3}{3} + \frac{2}{5 x y}$

Step 3.2.2

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

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Step 3.2.2.1

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

$\frac{4 x^{2}}{3 (x^{2} x^{1})} + \frac{8}{x^{3}} \cdot \frac{3}{3} + \frac{2}{5 x y}$

Step 3.2.2.2

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

$\frac{4 x^{2}}{3 x^{2 + 1}} + \frac{8}{x^{3}} \cdot \frac{3}{3} + \frac{2}{5 x y}$

Step 3.2.3

Add $2$ and $1$ .

$\frac{4 x^{2}}{3 x^{3}} + \frac{8}{x^{3}} \cdot \frac{3}{3} + \frac{2}{5 x y}$

Step 3.3

Multiply $\frac{8}{x^{3}}$ by $\frac{3}{3}$ .

$\frac{4 x^{2}}{3 x^{3}} + \frac{8 \cdot 3}{x^{3} \cdot 3} + \frac{2}{5 x y}$

Step 3.4

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

$\frac{4 x^{2}}{3 x^{3}} + \frac{8 \cdot 3}{3 x^{3}} + \frac{2}{5 x y}$

Step 4

Combine the numerators over the common denominator.

$\frac{4 x^{2} + 8 \cdot 3}{3 x^{3}} + \frac{2}{5 x y}$

Step 5

Multiply $8$ by $3$ .

$\frac{4 x^{2} + 24}{3 x^{3}} + \frac{2}{5 x y}$

Step 6

Factor $4$ out of $4 x^{2} + 24$ .

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Step 6.1

Factor $4$ out of $4 x^{2}$ .

$\frac{4 (x^{2}) + 24}{3 x^{3}} + \frac{2}{5 x y}$

Step 6.2

Factor $4$ out of $24$ .

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

Step 6.3

Factor $4$ out of $4 x^{2} + 4 \cdot 6$ .

$\frac{4 (x^{2} + 6)}{3 x^{3}} + \frac{2}{5 x y}$

Step 7

To write $\frac{4 (x^{2} + 6)}{3 x^{3}}$ as a fraction with a common denominator, multiply by $\frac{5 y}{5 y}$ .

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

Step 8

To write $\frac{2}{5 x y}$ as a fraction with a common denominator, multiply by $\frac{3 x^{2}}{3 x^{2}}$ .

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

Step 9

Write each expression with a common denominator of $15 y x^{3}$ , by multiplying each by an appropriate factor of $1$ .

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Step 9.1

Multiply $\frac{4 (x^{2} + 6)}{3 x^{3}}$ by $\frac{5 y}{5 y}$ .

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

Step 9.2

Multiply $5$ by $3$ .

$\frac{4 (x^{2} + 6) (5 y)}{15 x^{3} y} + \frac{2}{5 x y} \cdot \frac{3 x^{2}}{3 x^{2}}$

Step 9.3

Multiply $\frac{2}{5 x y}$ by $\frac{3 x^{2}}{3 x^{2}}$ .

$\frac{4 (x^{2} + 6) (5 y)}{15 x^{3} y} + \frac{2 (3 x^{2})}{5 x y (3 x^{2})}$

Step 9.4

Multiply $3$ by $5$ .

$\frac{4 (x^{2} + 6) (5 y)}{15 x^{3} y} + \frac{2 (3 x^{2})}{15 x y x^{2}}$

Step 9.5

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

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Step 9.5.1

Move $x^{2}$ .

$\frac{4 (x^{2} + 6) (5 y)}{15 x^{3} y} + \frac{2 (3 x^{2})}{15 (x^{2} x) y}$

Step 9.5.2

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

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Step 9.5.2.1

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

$\frac{4 (x^{2} + 6) (5 y)}{15 x^{3} y} + \frac{2 (3 x^{2})}{15 (x^{2} x^{1}) y}$

Step 9.5.2.2

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

$\frac{4 (x^{2} + 6) (5 y)}{15 x^{3} y} + \frac{2 (3 x^{2})}{15 x^{2 + 1} y}$

Step 9.5.3

Add $2$ and $1$ .

$\frac{4 (x^{2} + 6) (5 y)}{15 x^{3} y} + \frac{2 (3 x^{2})}{15 x^{3} y}$

Step 10

Combine the numerators over the common denominator.

$\frac{4 (x^{2} + 6) (5 y) + 2 (3 x^{2})}{15 x^{3} y}$

Step 11

Simplify the numerator.

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Step 11.1

Factor $2$ out of $4 (x^{2} + 6) \cdot 5 y + 2 \cdot 3 x^{2}$ .

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Step 11.1.1

Factor $2$ out of $4 (x^{2} + 6) \cdot 5 y$ .

$\frac{2 (2 (x^{2} + 6) \cdot 5 y) + 2 \cdot 3 x^{2}}{15 x^{3} y}$

Step 11.1.2

Factor $2$ out of $2 \cdot 3 x^{2}$ .

$\frac{2 (2 (x^{2} + 6) \cdot 5 y) + 2 (3 x^{2})}{15 x^{3} y}$

Step 11.1.3

Factor $2$ out of $2 (2 (x^{2} + 6) \cdot 5 y) + 2 (3 x^{2})$ .

$\frac{2 (2 (x^{2} + 6) \cdot 5 y + 3 x^{2})}{15 x^{3} y}$

Step 11.2

Apply the distributive property.

$\frac{2 ((2 x^{2} + 2 \cdot 6) \cdot 5 y + 3 x^{2})}{15 x^{3} y}$

Step 11.3

Multiply $2$ by $6$ .

$\frac{2 ((2 x^{2} + 12) \cdot 5 y + 3 x^{2})}{15 x^{3} y}$

Step 11.4

Apply the distributive property.

$\frac{2 ((2 x^{2} \cdot 5 + 12 \cdot 5) y + 3 x^{2})}{15 x^{3} y}$

Step 11.5

Multiply $5$ by $2$ .

$\frac{2 ((10 x^{2} + 12 \cdot 5) y + 3 x^{2})}{15 x^{3} y}$

Step 11.6

Multiply $12$ by $5$ .

$\frac{2 ((10 x^{2} + 60) y + 3 x^{2})}{15 x^{3} y}$

Step 11.7

Apply the distributive property.

$\frac{2 (10 x^{2} y + 60 y + 3 x^{2})}{15 x^{3} y}$