Algebra Examples

$\frac{1}{a^{2} + a b} + \frac{1}{a b + b^{2}}$

Step 1

Simplify each term.

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Step 1.1

Factor $a$ out of $a^{2} + a b$ .

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Step 1.1.1

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

$\frac{1}{a \cdot a + a b} + \frac{1}{a b + b^{2}}$

Step 1.1.2

Factor $a$ out of $a b$ .

$\frac{1}{a \cdot a + a (b)} + \frac{1}{a b + b^{2}}$

Step 1.1.3

Factor $a$ out of $a \cdot a + a (b)$ .

$\frac{1}{a (a + b)} + \frac{1}{a b + b^{2}}$

Step 1.2

Factor $b$ out of $a b + b^{2}$ .

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Step 1.2.1

Factor $b$ out of $a b$ .

$\frac{1}{a (a + b)} + \frac{1}{b a + b^{2}}$

Step 1.2.2

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

$\frac{1}{a (a + b)} + \frac{1}{b a + b \cdot b}$

Step 1.2.3

Factor $b$ out of $b a + b \cdot b$ .

$\frac{1}{a (a + b)} + \frac{1}{b (a + b)}$

Step 2

To write $\frac{1}{a (a + b)}$ as a fraction with a common denominator, multiply by $\frac{b}{b}$ .

$\frac{1}{a (a + b)} \cdot \frac{b}{b} + \frac{1}{b (a + b)}$

Step 3

To write $\frac{1}{b (a + b)}$ as a fraction with a common denominator, multiply by $\frac{a}{a}$ .

$\frac{1}{a (a + b)} \cdot \frac{b}{b} + \frac{1}{b (a + b)} \cdot \frac{a}{a}$

Step 4

Write each expression with a common denominator of $a (a + b) b$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{1}{a (a + b)}$ by $\frac{b}{b}$ .

$\frac{b}{a (a + b) b} + \frac{1}{b (a + b)} \cdot \frac{a}{a}$

Step 4.2

Multiply $\frac{1}{b (a + b)}$ by $\frac{a}{a}$ .

$\frac{b}{a (a + b) b} + \frac{a}{b (a + b) a}$

Step 4.3

Reorder the factors of $a (a + b) b$ .

$\frac{b}{a b (a + b)} + \frac{a}{b (a + b) a}$

Step 4.4

Reorder the factors of $b (a + b) a$ .

$\frac{b}{a b (a + b)} + \frac{a}{a b (a + b)}$

Step 5

Combine the numerators over the common denominator.

$\frac{b + a}{a b (a + b)}$

Step 6

Cancel the common factor of $b + a$ and $a + b$ .

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Step 6.1

Reorder terms.

$\frac{a + b}{a b (a + b)}$

Step 6.2

Cancel the common factor.

$\frac{a + b}{a b (a + b)}$

Step 6.3

Rewrite the expression.

$\frac{1}{a b}$