Algebra Examples

$(\frac{a}{b} + \frac{1}{a}) (a - \frac{a^{3}}{a^{2} + b})$

Step 1

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

$(\frac{a}{b} + \frac{1}{a}) (\frac{a (a^{2} + b)}{a^{2} + b} - \frac{a^{3}}{a^{2} + b})$

Step 2

Combine the numerators over the common denominator.

$(\frac{a}{b} + \frac{1}{a}) \frac{a (a^{2} + b) - a^{3}}{a^{2} + b}$

Step 3

Simplify the numerator.

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Step 3.1

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

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Step 3.1.1

Factor $a$ out of $- a^{3}$ .

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

Step 3.1.2

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

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

Step 3.2

Subtract $a^{2}$ from $a^{2}$ .

$(\frac{a}{b} + \frac{1}{a}) \frac{a (b + 0)}{a^{2} + b}$

Step 3.3

Add $b$ and $0$ .

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

Step 4

Multiply $\frac{a}{b} + \frac{1}{a}$ by $\frac{a b}{a^{2} + b}$ .

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

Step 5

Simplify the numerator.

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Step 5.1

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

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

Step 5.2

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

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

Step 5.3

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

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Step 5.3.1

Multiply $\frac{a}{b}$ by $\frac{a}{a}$ .

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

Step 5.3.2

Multiply $\frac{1}{a}$ by $\frac{b}{b}$ .

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

Step 5.3.3

Reorder the factors of $b a$ .

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

Step 5.4

Combine the numerators over the common denominator.

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

Step 5.5

Multiply $a$ by $a$ .

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

Step 5.6

Combine exponents.

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Step 5.6.1

Combine $\frac{a^{2} + b}{a b}$ and $a$ .

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

Step 5.6.2

Combine $\frac{(a^{2} + b) a}{a b}$ and $b$ .

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

Step 5.7

Reduce the expression $\frac{(a^{2} + b) a b}{a b}$ by cancelling the common factors.

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Step 5.7.1

Cancel the common factor.

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

Step 5.7.2

Rewrite the expression.

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

Step 5.8

Cancel the common factor of $b$ .

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Step 5.8.1

Cancel the common factor.

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

Step 5.8.2

Divide $a^{2} + b$ by $1$ .

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

Step 6

Cancel the common factor of $a^{2} + b$ .

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Step 6.1

Cancel the common factor.

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

Step 6.2

Rewrite the expression.

$1$