Basic Math Examples

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

Step 1

Apply the product rule to $c d$ .

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

Step 2

Move all terms containing $c$ to the left side of the equation.

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Step 2.1

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

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

Step 2.2

To write $- c^{8} d^{8}$ as a fraction with a common denominator, multiply by $\frac{c d - r}{c d - r}$ .

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

Step 2.3

Combine $- c^{8} d^{8}$ and $\frac{c d - r}{c d - r}$ .

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

Step 2.4

Combine the numerators over the common denominator.

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

Step 2.5

Simplify the numerator.

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Step 2.5.1

Apply the distributive property.

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

Step 2.5.2

Multiply $c^{8}$ by $c$ by adding the exponents.

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Step 2.5.2.1

Move $c$ .

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

Step 2.5.2.2

Multiply $c$ by $c^{8}$ .

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Step 2.5.2.2.1

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

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

Step 2.5.2.2.2

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

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

Step 2.5.2.3

Add $1$ and $8$ .

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

Step 2.5.3

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

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Step 2.5.3.1

Multiply $- 1$ by $- 1$ .

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

Step 2.5.3.2

Multiply $c^{8}$ by $1$ .

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

Step 2.5.4

Multiply $d^{8}$ by $d$ by adding the exponents.

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Step 2.5.4.1

Move $d$ .

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

Step 2.5.4.2

Multiply $d$ by $d^{8}$ .

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Step 2.5.4.2.1

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

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

Step 2.5.4.2.2

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

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

Step 2.5.4.3

Add $1$ and $8$ .

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