Basic Math Examples

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

Step 1

Multiply both sides by $c d c d$ .

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

Step 2

Simplify.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Simplify $\frac{a b a b}{c d c d} (c d c d)$ .

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Step 2.1.1.1

Cancel the common factor of $c^{3} d^{3}$ .

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Step 2.1.1.1.1

Factor $a b$ out of $a b a b$ .

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

Step 2.1.1.1.2

Factor $c^{3} d^{3}$ out of $c d c d$ .

$\frac{a b (a b)}{c^{3} d^{3} (c^{- 2} d^{- 2} c d)} (c d c d) = \frac{a b}{c d} (c d c d)$

Step 2.1.1.1.3

Factor $c^{3} d^{3}$ out of $c d c d$ .

$\frac{a b (a b)}{c^{3} d^{3} (c^{- 2} d^{- 2} c d)} (c^{3} d^{3} (c^{- 2} d^{- 2} c d)) = \frac{a b}{c d} (c d c d)$

Step 2.1.1.1.4

Cancel the common factor.

$\frac{a b (a b)}{c^{3} d^{3} (c^{- 2} d^{- 2} c d)} (c^{3} d^{3} (c^{- 2} d^{- 2} c d)) = \frac{a b}{c d} (c d c d)$

Step 2.1.1.1.5

Rewrite the expression.

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

Step 2.1.1.2

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

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

Step 2.1.1.3

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

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

Step 2.1.1.4

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

$\frac{a^{1 + 1} b \cdot b}{c^{- 2} d^{- 2} c d} (c^{- 2} d^{- 2} c d) = \frac{a b}{c d} (c d c d)$

Step 2.1.1.5

Add $1$ and $1$ .

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

Step 2.1.1.6

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

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

Step 2.1.1.7

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

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

Step 2.1.1.8

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

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

Step 2.1.1.9

Add $1$ and $1$ .

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

Step 2.1.1.10

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

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

Step 2.1.1.11

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

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

Step 2.1.1.12

Subtract $2$ from $1$ .

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

Step 2.1.1.13

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

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

Step 2.1.1.14

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

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

Step 2.1.1.15

Subtract $2$ from $1$ .

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

Step 2.1.1.16

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

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

Step 2.1.1.17

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

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

Step 2.1.1.18

Subtract $2$ from $1$ .

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

Step 2.1.1.19

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

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

Step 2.1.1.20

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

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

Step 2.1.1.21

Subtract $2$ from $1$ .

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

Step 2.1.1.22

Combine $c^{- 1}$ and $\frac{a^{2} b^{2}}{c^{- 1} d^{- 1}}$ .

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

Step 2.1.1.23

Combine $\frac{c^{- 1} (a^{2} b^{2})}{c^{- 1} d^{- 1}}$ and $d^{- 1}$ .

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

Step 2.1.1.24

Cancel the common factor of $c^{- 1}$ .

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Step 2.1.1.24.1

Cancel the common factor.

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

Step 2.1.1.24.2

Rewrite the expression.

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

Step 2.1.1.25

Cancel the common factor of $d^{- 1}$ .

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Step 2.1.1.25.1

Cancel the common factor.

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

Step 2.1.1.25.2

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

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

Step 2.2

Simplify the right side.

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Step 2.2.1

Simplify $\frac{a b}{c d} (c d c d)$ .

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Step 2.2.1.1

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

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Step 2.2.1.1.1

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

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

Step 2.2.1.1.2

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

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

Step 2.2.1.1.3

Cancel the common factor.

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

Step 2.2.1.1.4

Rewrite the expression.

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

Step 2.2.1.2

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

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

Step 2.2.1.3

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

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

Step 2.2.1.4

Subtract $1$ from $1$ .

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

Step 2.2.1.5

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

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

Step 2.2.1.6

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

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

Step 2.2.1.7

Subtract $1$ from $1$ .

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

Step 2.2.1.8

Combine $c^{0}$ and $\frac{a b}{c^{- 1} d^{- 1}}$ .

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

Step 2.2.1.9

Combine $\frac{c^{0} (a b)}{c^{- 1} d^{- 1}}$ and $d^{0}$ .

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

Step 2.2.1.10

Raise to zero.

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Step 2.2.1.10.1

Move $c^{- 1}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

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

Step 2.2.1.10.2

Move $d^{- 1}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

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

Step 2.2.1.11

Multiply $c^{0}$ by $c$ by adding the exponents.

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Step 2.2.1.11.1

Move $c$ .

$a^{2} b^{2} = c \cdot c^{0} (a b) d^{0} d$

Step 2.2.1.11.2

Multiply $c$ by $c^{0}$ .

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Step 2.2.1.11.2.1

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

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

Step 2.2.1.11.2.2

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

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

Step 2.2.1.11.3

Add $1$ and $0$ .

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

Step 2.2.1.12

Simplify $c^{1} (a b) d^{0}$ .

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

Step 3

Solve for $a$ .

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Step 3.1

Subtract $c a b d$ from both sides of the equation.

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

Step 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

Factor $a b$ out of $- c a b d$ .

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

Step 3.2.3

Factor $a b$ out of $a b (a b) + a b (- c d)$ .

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

Step 3.3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$a = 0$

$a b - c d = 0$

Step 3.4

Set $a$ equal to $0$ .

$a = 0$

Step 3.5

Set $a b - c d$ equal to $0$ and solve for $a$ .

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Step 3.5.1

Set $a b - c d$ equal to $0$ .

$a b - c d = 0$

Step 3.5.2

Solve $a b - c d = 0$ for $a$ .

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Step 3.5.2.1

Add $c d$ to both sides of the equation.

$a b = c d$

Step 3.5.2.2

Divide each term in $a b = c d$ by $b$ and simplify.

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Step 3.5.2.2.1

Divide each term in $a b = c d$ by $b$ .

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

Step 3.5.2.2.2

Simplify the left side.

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Step 3.5.2.2.2.1

Cancel the common factor of $b$ .

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Step 3.5.2.2.2.1.1

Cancel the common factor.

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

Step 3.5.2.2.2.1.2

Divide $a$ by $1$ .

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

Step 3.6

The final solution is all the values that make $a b (a b - c d) = 0$ true.

$a = 0$

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

$a = 0$

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