Algebra Examples

$\frac{a b + b^{2}}{3} \div \frac{b^{3}}{3 a} + \frac{a + b}{b}$

Step 1

Simplify each term.

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Step 1.1

To divide by a fraction, multiply by its reciprocal.

$\frac{a b + b^{2}}{3} \cdot \frac{3 a}{b^{3}} + \frac{a + b}{b}$

Step 1.2

Cancel the common factor of $3$ .

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Step 1.2.1

Factor $3$ out of $3 a$ .

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

Step 1.2.2

Cancel the common factor.

$\frac{a b + b^{2}}{3} \cdot \frac{3 a}{b^{3}} + \frac{a + b}{b}$

Step 1.2.3

Rewrite the expression.

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

Step 1.3

Multiply $a b + b^{2}$ by $\frac{a}{b^{3}}$ .

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

Step 1.4

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

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Step 1.4.1

Factor $b$ out of $a b$ .

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

Step 1.4.2

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

$\frac{(b a + b \cdot b) a}{b^{3}} + \frac{a + b}{b}$

Step 1.4.3

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

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

Step 1.5

Cancel the common factor of $b$ and $b^{3}$ .

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Step 1.5.1

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

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

Step 1.5.2

Cancel the common factors.

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Step 1.5.2.1

Factor $b$ out of $b^{3}$ .

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

Step 1.5.2.2

Cancel the common factor.

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

Step 1.5.2.3

Rewrite the expression.

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

Step 2

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

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

Step 3

Write each expression with a common denominator of $b^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

Add $1$ and $1$ .

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

Step 4

Simplify terms.

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Step 4.1

Combine the numerators over the common denominator.

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

Step 4.2

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

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Step 4.2.1

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

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

Step 4.2.2

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

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

Step 4.2.3

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

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

Step 5

Simplify the numerator.

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Step 5.1

Raise $a + b$ to the power of $1$ .

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

Step 5.2

Raise $a + b$ to the power of $1$ .

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

Step 5.3

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

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

Step 5.4

Add $1$ and $1$ .

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