Basic Math Examples

$5 + \frac{5 + \frac{1}{b}}{\frac{1}{b^{2}} + \frac{5}{b}}$

Step 1

To write $5$ as a fraction with a common denominator, multiply by $\frac{b}{b}$ .

$5 + \frac{\frac{5 b}{b} + \frac{1}{b}}{\frac{1}{b^{2}} + \frac{5}{b}}$

Step 2

Combine the numerators over the common denominator.

$5 + \frac{\frac{5 b + 1}{b}}{\frac{1}{b^{2}} + \frac{5}{b}}$

Step 3

Simplify each term.

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Step 3.1

Multiply the numerator by the reciprocal of the denominator.

$5 + \frac{5 b + 1}{b} \cdot \frac{1}{\frac{1}{b^{2}} + \frac{5}{b}}$

Step 3.2

Simplify the denominator.

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Step 3.2.1

To write $\frac{5}{b}$ as a fraction with a common denominator, multiply by $\frac{b}{b}$ .

$5 + \frac{5 b + 1}{b} \cdot \frac{1}{\frac{1}{b^{2}} + \frac{5}{b} \cdot \frac{b}{b}}$

Step 3.2.2

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

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Step 3.2.2.1

Multiply $\frac{5}{b}$ by $\frac{b}{b}$ .

$5 + \frac{5 b + 1}{b} \cdot \frac{1}{\frac{1}{b^{2}} + \frac{5 b}{b \cdot b}}$

Step 3.2.2.2

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

$5 + \frac{5 b + 1}{b} \cdot \frac{1}{\frac{1}{b^{2}} + \frac{5 b}{b^{1} b}}$

Step 3.2.2.3

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

$5 + \frac{5 b + 1}{b} \cdot \frac{1}{\frac{1}{b^{2}} + \frac{5 b}{b^{1} b^{1}}}$

Step 3.2.2.4

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

$5 + \frac{5 b + 1}{b} \cdot \frac{1}{\frac{1}{b^{2}} + \frac{5 b}{b^{1 + 1}}}$

Step 3.2.2.5

Add $1$ and $1$ .

$5 + \frac{5 b + 1}{b} \cdot \frac{1}{\frac{1}{b^{2}} + \frac{5 b}{b^{2}}}$

Step 3.2.3

Combine the numerators over the common denominator.

$5 + \frac{5 b + 1}{b} \cdot \frac{1}{\frac{1 + 5 b}{b^{2}}}$

Step 3.3

Multiply the numerator by the reciprocal of the denominator.

$5 + \frac{5 b + 1}{b} (1 \frac{b^{2}}{1 + 5 b})$

Step 3.4

Multiply $\frac{b^{2}}{1 + 5 b}$ by $1$ .

$5 + \frac{5 b + 1}{b} \cdot \frac{b^{2}}{1 + 5 b}$

Step 3.5

Cancel the common factor of $b$ .

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Step 3.5.1

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

$5 + \frac{5 b + 1}{b} \cdot \frac{b \cdot b}{1 + 5 b}$

Step 3.5.2

Cancel the common factor.

$5 + \frac{5 b + 1}{b} \cdot \frac{b \cdot b}{1 + 5 b}$

Step 3.5.3

Rewrite the expression.

$5 + (5 b + 1) \frac{b}{1 + 5 b}$

Step 3.6

Multiply $5 b + 1$ by $\frac{b}{1 + 5 b}$ .

$5 + \frac{(5 b + 1) b}{1 + 5 b}$

Step 3.7

Cancel the common factor of $5 b + 1$ and $1 + 5 b$ .

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Step 3.7.1

Reorder terms.

$5 + \frac{(5 b + 1) b}{5 b + 1}$

Step 3.7.2

Cancel the common factor.

$5 + \frac{(5 b + 1) b}{5 b + 1}$

Step 3.7.3

Divide $b$ by $1$ .

$5 + b$