Basic Math Examples

\frac{{(c d)}^{4 r}}{(c d) - r} = c^{8} d^{8}

$\frac{{(c d)}^{4 r}}{(c d) - r} = c^{8} d^{8}$

Step 1

Apply the product rule to

c d

$c d$ .

\frac{c^{4 r} d^{4 r}}{c d - r} = c^{8} d^{8}

$\frac{c^{4 r} d^{4 r}}{c d - r} = c^{8} d^{8}$

Step 2

Move all terms containing

c

$c$ to the left side of the equation.

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Step 2.1

Subtract

c^{8} d^{8}

$c^{8} d^{8}$ from both sides of the equation.

\frac{c^{4 r} d^{4 r}}{c d - r} - c^{8} d^{8} = 0

$\frac{c^{4 r} d^{4 r}}{c d - r} - c^{8} d^{8} = 0$

Step 2.2

To write

- c^{8} d^{8}

$- c^{8} d^{8}$ as a fraction with a common denominator, multiply by

\frac{c d - r}{c d - r}

$\frac{c d - r}{c d - r}$ .

\frac{c^{4 r} d^{4 r}}{c d - r} - c^{8} d^{8} \cdot \frac{c d - r}{c d - r} = 0

$\frac{c^{4 r} d^{4 r}}{c d - r} - c^{8} d^{8} \cdot \frac{c d - r}{c d - r} = 0$

Step 2.3

Combine

- c^{8} d^{8}

$- c^{8} d^{8}$ and

\frac{c d - r}{c d - r}

$\frac{c d - r}{c d - r}$ .

\frac{c^{4 r} d^{4 r}}{c d - r} + \frac{- c^{8} d^{8} (c d - r)}{c d - r} = 0

$\frac{c^{4 r} d^{4 r}}{c d - r} + \frac{- c^{8} d^{8} (c d - r)}{c d - r} = 0$

Step 2.4

Combine the numerators over the common denominator.

\frac{c^{4 r} d^{4 r} - c^{8} d^{8} (c d - r)}{c d - r} = 0

$\frac{c^{4 r} d^{4 r} - c^{8} d^{8} (c d - r)}{c d - r} = 0$

Step 2.5

Simplify the numerator.

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Step 2.5.1

Apply the distributive property.

\frac{c^{4 r} d^{4 r} - c^{8} d^{8} (c d) - c^{8} d^{8} (- r)}{c d - r} = 0

$\frac{c^{4 r} d^{4 r} - c^{8} d^{8} (c d) - c^{8} d^{8} (- r)}{c d - r} = 0$

Step 2.5.2

Multiply

c^{8}

$c^{8}$ by

c

$c$ by adding the exponents.

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Step 2.5.2.1

Move

c

$c$ .

\frac{c^{4 r} d^{4 r} - (c \cdot c^{8}) d^{8} d - c^{8} d^{8} (- r)}{c d - r} = 0

$\frac{c^{4 r} d^{4 r} - (c \cdot c^{8}) d^{8} d - c^{8} d^{8} (- r)}{c d - r} = 0$

Step 2.5.2.2

Multiply

c

$c$ by

c^{8}

$c^{8}$ .

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Step 2.5.2.2.1

Raise

c

$c$ to the power of

1

$1$ .

\frac{c^{4 r} d^{4 r} - (c^{1} c^{8}) d^{8} d - c^{8} d^{8} (- r)}{c d - r} = 0

$\frac{c^{4 r} d^{4 r} - (c^{1} c^{8}) d^{8} d - c^{8} d^{8} (- r)}{c d - r} = 0$

Step 2.5.2.2.2

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

\frac{c^{4 r} d^{4 r} - c^{1 + 8} d^{8} d - c^{8} d^{8} (- r)}{c d - r} = 0

$\frac{c^{4 r} d^{4 r} - c^{1 + 8} d^{8} d - c^{8} d^{8} (- r)}{c d - r} = 0$

\frac{c^{4 r} d^{4 r} - c^{1 + 8} d^{8} d - c^{8} d^{8} (- r)}{c d - r} = 0

$\frac{c^{4 r} d^{4 r} - c^{1 + 8} d^{8} d - c^{8} d^{8} (- r)}{c d - r} = 0$

Step 2.5.2.3

Add

1

$1$ and

8

$8$ .

\frac{c^{4 r} d^{4 r} - c^{9} d^{8} d - c^{8} d^{8} (- r)}{c d - r} = 0

$\frac{c^{4 r} d^{4 r} - c^{9} d^{8} d - c^{8} d^{8} (- r)}{c d - r} = 0$

\frac{c^{4 r} d^{4 r} - c^{9} d^{8} d - c^{8} d^{8} (- r)}{c d - r} = 0

$\frac{c^{4 r} d^{4 r} - c^{9} d^{8} d - c^{8} d^{8} (- r)}{c d - r} = 0$

Step 2.5.3

Multiply

- c^{8} d^{8} (- r)

$- c^{8} d^{8} (- r)$ .

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Step 2.5.3.1

Multiply

- 1

$- 1$ by

- 1

$- 1$ .

\frac{c^{4 r} d^{4 r} - c^{9} d^{8} d + 1 c^{8} d^{8} r}{c d - r} = 0

$\frac{c^{4 r} d^{4 r} - c^{9} d^{8} d + 1 c^{8} d^{8} r}{c d - r} = 0$

Step 2.5.3.2

Multiply

c^{8}

$c^{8}$ by

1

$1$ .

\frac{c^{4 r} d^{4 r} - c^{9} d^{8} d + c^{8} d^{8} r}{c d - r} = 0

$\frac{c^{4 r} d^{4 r} - c^{9} d^{8} d + c^{8} d^{8} r}{c d - r} = 0$

\frac{c^{4 r} d^{4 r} - c^{9} d^{8} d + c^{8} d^{8} r}{c d - r} = 0

$\frac{c^{4 r} d^{4 r} - c^{9} d^{8} d + c^{8} d^{8} r}{c d - r} = 0$

Step 2.5.4

Multiply

d^{8}

$d^{8}$ by

d

$d$ by adding the exponents.

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Step 2.5.4.1

Move

d

$d$ .

\frac{c^{4 r} d^{4 r} - c^{9} (d \cdot d^{8}) + c^{8} d^{8} r}{c d - r} = 0

$\frac{c^{4 r} d^{4 r} - c^{9} (d \cdot d^{8}) + c^{8} d^{8} r}{c d - r} = 0$

Step 2.5.4.2

Multiply

d

$d$ by

d^{8}

$d^{8}$ .

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Step 2.5.4.2.1

Raise

d

$d$ to the power of

1

$1$ .

\frac{c^{4 r} d^{4 r} - c^{9} (d^{1} d^{8}) + c^{8} d^{8} r}{c d - r} = 0

$\frac{c^{4 r} d^{4 r} - c^{9} (d^{1} d^{8}) + c^{8} d^{8} r}{c d - r} = 0$

Step 2.5.4.2.2

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

\frac{c^{4 r} d^{4 r} - c^{9} d^{1 + 8} + c^{8} d^{8} r}{c d - r} = 0

$\frac{c^{4 r} d^{4 r} - c^{9} d^{1 + 8} + c^{8} d^{8} r}{c d - r} = 0$

\frac{c^{4 r} d^{4 r} - c^{9} d^{1 + 8} + c^{8} d^{8} r}{c d - r} = 0

$\frac{c^{4 r} d^{4 r} - c^{9} d^{1 + 8} + c^{8} d^{8} r}{c d - r} = 0$

Step 2.5.4.3

Add

1

$1$ and

8

$8$ .

\frac{c^{4 r} d^{4 r} - c^{9} d^{9} + c^{8} d^{8} r}{c d - r} = 0

$\frac{c^{4 r} d^{4 r} - c^{9} d^{9} + c^{8} d^{8} r}{c d - r} = 0$

\frac{c^{4 r} d^{4 r} - c^{9} d^{9} + c^{8} d^{8} r}{c d - r} = 0

$\frac{c^{4 r} d^{4 r} - c^{9} d^{9} + c^{8} d^{8} r}{c d - r} = 0$

\frac{c^{4 r} d^{4 r} - c^{9} d^{9} + c^{8} d^{8} r}{c d - r} = 0

$\frac{c^{4 r} d^{4 r} - c^{9} d^{9} + c^{8} d^{8} r}{c d - r} = 0$

\frac{c^{4 r} d^{4 r} - c^{9} d^{9} + c^{8} d^{8} r}{c d - r} = 0

$\frac{c^{4 r} d^{4 r} - c^{9} d^{9} + c^{8} d^{8} r}{c d - r} = 0$