Pre-Algebra Examples

Add: $\frac{10}{x^{2} + 3 x} + \frac{15}{x^{2}}$

Step 1

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

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Step 1.1

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

$\frac{10}{x \cdot x + 3 x} + \frac{15}{x^{2}}$

Step 1.2

Factor $x$ out of $3 x$ .

$\frac{10}{x \cdot x + x \cdot 3} + \frac{15}{x^{2}}$

Step 1.3

Factor $x$ out of $x \cdot x + x \cdot 3$ .

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

Step 2

To write $\frac{10}{x (x + 3)}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

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

Step 3

To write $\frac{15}{x^{2}}$ as a fraction with a common denominator, multiply by $\frac{x + 3}{x + 3}$ .

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

Step 4

Write each expression with a common denominator of $(x + 3) x^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{10}{x (x + 3)}$ by $\frac{x}{x}$ .

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

Step 4.2

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

$\frac{10 x}{x^{1} x (x + 3)} + \frac{15}{x^{2}} \cdot \frac{x + 3}{x + 3}$

Step 4.3

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

$\frac{10 x}{x^{1} x^{1} (x + 3)} + \frac{15}{x^{2}} \cdot \frac{x + 3}{x + 3}$

Step 4.4

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

$\frac{10 x}{x^{1 + 1} (x + 3)} + \frac{15}{x^{2}} \cdot \frac{x + 3}{x + 3}$

Step 4.5

Add $1$ and $1$ .

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

Step 4.6

Multiply $\frac{15}{x^{2}}$ by $\frac{x + 3}{x + 3}$ .

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Factor $5$ out of $10 x + 15 (x + 3)$ .

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Step 6.1.1

Factor $5$ out of $10 x$ .

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

Step 6.1.2

Factor $5$ out of $15 (x + 3)$ .

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

Step 6.1.3

Factor $5$ out of $5 (2 x) + 5 (3 (x + 3))$ .

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

Step 6.2

Apply the distributive property.

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

Step 6.3

Multiply $3$ by $3$ .

$\frac{5 (2 x + 3 x + 9)}{x^{2} (x + 3)}$

Step 6.4

Add $2 x$ and $3 x$ .

$\frac{5 (5 x + 9)}{x^{2} (x + 3)}$