Basic Math Examples

$\frac{a}{\frac{b}{\frac{c}{d}}} = \frac{a}{b} \cdot \frac{d}{c}$

Step 1

Multiply both sides by $\frac{b}{\frac{c}{d}}$ .

$\frac{a}{\frac{b}{\frac{c}{d}}} \cdot \frac{b}{\frac{c}{d}} = \frac{a}{b} \cdot \frac{d}{c} \frac{b}{\frac{c}{d}}$

Step 2

Simplify.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Simplify $\frac{a}{\frac{b}{\frac{c}{d}}} \cdot \frac{b}{\frac{c}{d}}$ .

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Step 2.1.1.1

Multiply the numerator by the reciprocal of the denominator.

$a \frac{\frac{c}{d}}{b} \frac{b}{\frac{c}{d}} = \frac{a}{b} \cdot \frac{d}{c} \frac{b}{\frac{c}{d}}$

Step 2.1.1.2

Multiply the numerator by the reciprocal of the denominator.

$a (\frac{c}{d} \cdot \frac{1}{b}) \frac{b}{\frac{c}{d}} = \frac{a}{b} \cdot \frac{d}{c} \frac{b}{\frac{c}{d}}$

Step 2.1.1.3

Cancel the common factor of $\frac{c}{d}$ .

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Step 2.1.1.3.1

Factor $\frac{c}{d}$ out of $a (\frac{c}{d} \cdot \frac{1}{b})$ .

$\frac{c}{d} (a (\frac{1}{b})) \frac{b}{\frac{c}{d}} = \frac{a}{b} \cdot \frac{d}{c} \frac{b}{\frac{c}{d}}$

Step 2.1.1.3.2

Cancel the common factor.

$\frac{c}{d} (a (\frac{1}{b})) \frac{b}{\frac{c}{d}} = \frac{a}{b} \cdot \frac{d}{c} \frac{b}{\frac{c}{d}}$

Step 2.1.1.3.3

Rewrite the expression.

$a (\frac{1}{b}) b = \frac{a}{b} \cdot \frac{d}{c} \frac{b}{\frac{c}{d}}$

Step 2.1.1.4

Combine $a$ and $\frac{1}{b}$ .

$\frac{a}{b} b = \frac{a}{b} \cdot \frac{d}{c} \frac{b}{\frac{c}{d}}$

Step 2.1.1.5

Combine $\frac{a}{b}$ and $b$ .

$\frac{a b}{b} = \frac{a}{b} \cdot \frac{d}{c} \frac{b}{\frac{c}{d}}$

Step 2.1.1.6

Cancel the common factor of $b$ .

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Step 2.1.1.6.1

Cancel the common factor.

$\frac{a b}{b} = \frac{a}{b} \cdot \frac{d}{c} \frac{b}{\frac{c}{d}}$

Step 2.1.1.6.2

Divide $a$ by $1$ .

$a = \frac{a}{b} \cdot \frac{d}{c} \frac{b}{\frac{c}{d}}$

Step 2.2

Simplify the right side.

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Step 2.2.1

Simplify $\frac{a}{b} \cdot \frac{d}{c} \frac{b}{\frac{c}{d}}$ .

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Step 2.2.1.1

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

$a = \frac{a d}{b c} \cdot \frac{b}{\frac{c}{d}}$

Step 2.2.1.2

Cancel the common factor of $b$ .

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Step 2.2.1.2.1

Factor $b$ out of $b c$ .

$a = \frac{a d}{b (c)} \cdot \frac{b}{\frac{c}{d}}$

Step 2.2.1.2.2

Cancel the common factor.

$a = \frac{a d}{b c} \cdot \frac{b}{\frac{c}{d}}$

Step 2.2.1.2.3

Rewrite the expression.

$a = \frac{a d}{c} \cdot \frac{1}{\frac{c}{d}}$

Step 2.2.1.3

Multiply $\frac{a d}{c}$ by $\frac{1}{\frac{c}{d}}$ .

$a = \frac{a d}{c \frac{c}{d}}$

Step 2.2.1.4

Combine $c$ and $\frac{c}{d}$ .

$a = \frac{a d}{\frac{c \cdot c}{d}}$

Step 2.2.1.5

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

$a = \frac{a d}{\frac{c^{1} c}{d}}$

Step 2.2.1.6

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

$a = \frac{a d}{\frac{c^{1} c^{1}}{d}}$

Step 2.2.1.7

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

$a = \frac{a d}{\frac{c^{1 + 1}}{d}}$

Step 2.2.1.8

Add $1$ and $1$ .

$a = \frac{a d}{\frac{c^{2}}{d}}$

Step 2.2.1.9

Multiply the numerator by the reciprocal of the denominator.

$a = a d \frac{d}{c^{2}}$

Step 2.2.1.10

Multiply $a d \frac{d}{c^{2}}$ .

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Step 2.2.1.10.1

Combine $a$ and $\frac{d}{c^{2}}$ .

$a = d \frac{a d}{c^{2}}$

Step 2.2.1.10.2

Combine $d$ and $\frac{a d}{c^{2}}$ .

$a = \frac{d (a d)}{c^{2}}$

Step 2.2.1.10.3

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

$a = \frac{a (d^{1} d)}{c^{2}}$

Step 2.2.1.10.4

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

$a = \frac{a (d^{1} d^{1})}{c^{2}}$

Step 2.2.1.10.5

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

$a = \frac{a d^{1 + 1}}{c^{2}}$

Step 2.2.1.10.6

Add $1$ and $1$ .

$a = \frac{a d^{2}}{c^{2}}$

Step 3

Solve for $a$ .

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Step 3.1

Multiply both sides by $c^{2}$ .

$a c^{2} = \frac{a d^{2}}{c^{2}} c^{2}$

Step 3.2

Simplify the right side.

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Step 3.2.1

Cancel the common factor of $c^{2}$ .

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Step 3.2.1.1

Cancel the common factor.

$a c^{2} = \frac{a d^{2}}{c^{2}} c^{2}$

Step 3.2.1.2

Rewrite the expression.

$a c^{2} = a d^{2}$

Step 3.3

Solve for $a$ .

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Step 3.3.1

Subtract $a d^{2}$ from both sides of the equation.

$a c^{2} - a d^{2} = 0$

Step 3.3.2

Factor $a$ out of $a c^{2} - a d^{2}$ .

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Step 3.3.2.1

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

$a (c^{2}) - a d^{2} = 0$

Step 3.3.2.2

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

$a (c^{2}) + a (- 1 d^{2}) = 0$

Step 3.3.2.3

Factor $a$ out of $a (c^{2}) + a (- 1 d^{2})$ .

$a (c^{2} - 1 d^{2}) = 0$

Step 3.3.3

Factor.

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Step 3.3.3.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = c$ and $b = d$ .

$a ((c + d) (c - d)) = 0$

Step 3.3.3.2

Remove unnecessary parentheses.

$a (c + d) (c - d) = 0$

Step 3.3.4

Divide each term in $a (c + d) (c - d) = 0$ by $(c + d) (c - d)$ and simplify.

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Step 3.3.4.1

Divide each term in $a (c + d) (c - d) = 0$ by $(c + d) (c - d)$ .

$\frac{a (c + d) (c - d)}{(c + d) (c - d)} = \frac{0}{(c + d) (c - d)}$

Step 3.3.4.2

Simplify the left side.

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Step 3.3.4.2.1

Cancel the common factor of $c + d$ .

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Step 3.3.4.2.1.1

Cancel the common factor.

$\frac{a (c + d) (c - d)}{(c + d) (c - d)} = \frac{0}{(c + d) (c - d)}$

Step 3.3.4.2.1.2

Rewrite the expression.

$\frac{a (c - d)}{c - d} = \frac{0}{(c + d) (c - d)}$

Step 3.3.4.2.2

Cancel the common factor of $c - d$ .

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Step 3.3.4.2.2.1

Cancel the common factor.

$\frac{a (c - d)}{c - d} = \frac{0}{(c + d) (c - d)}$

Step 3.3.4.2.2.2

Divide $a$ by $1$ .

$a = \frac{0}{(c + d) (c - d)}$

Step 3.3.4.3

Simplify the right side.

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Step 3.3.4.3.1

Divide $0$ by $(c + d) (c - d)$ .

$a = 0$