Algebra Examples

$5 c^{- 7} d^{2} \cdot {(- c d^{2})}^{4}$

Step 1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$5 \frac{1}{c^{7}} d^{2} \cdot {(- c d^{2})}^{4}$

Step 2

Combine $5$ and $\frac{1}{c^{7}}$ .

$\frac{5}{c^{7}} d^{2} \cdot {(- c d^{2})}^{4}$

Step 3

Combine $\frac{5}{c^{7}}$ and $d^{2}$ .

$\frac{5 d^{2}}{c^{7}} \cdot {(- c d^{2})}^{4}$

Step 4

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 4.1

Apply the product rule to $- c d^{2}$ .

$\frac{5 d^{2}}{c^{7}} \cdot ({(- c)}^{4} {(d^{2})}^{4})$

Step 4.2

Apply the product rule to $- c$ .

$\frac{5 d^{2}}{c^{7}} \cdot ({(- 1)}^{4} c^{4} {(d^{2})}^{4})$

Step 5

Simplify terms.

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Step 5.1

Rewrite using the commutative property of multiplication.

${(- 1)}^{4} \frac{5 d^{2}}{c^{7}} (c^{4} {(d^{2})}^{4})$

Step 5.2

Simplify the expression.

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Step 5.2.1

Raise $- 1$ to the power of $4$ .

$1 \frac{5 d^{2}}{c^{7}} (c^{4} {(d^{2})}^{4})$

Step 5.2.2

Multiply $\frac{5 d^{2}}{c^{7}}$ by $1$ .

$\frac{5 d^{2}}{c^{7}} (c^{4} {(d^{2})}^{4})$

Step 5.3

Cancel the common factor of $c^{4}$ .

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Step 5.3.1

Factor $c^{4}$ out of $c^{7}$ .

$\frac{5 d^{2}}{c^{4} c^{3}} (c^{4} {(d^{2})}^{4})$

Step 5.3.2

Factor $c^{4}$ out of $c^{4} {(d^{2})}^{4}$ .

$\frac{5 d^{2}}{c^{4} c^{3}} (c^{4} ({(d^{2})}^{4}))$

Step 5.3.3

Cancel the common factor.

$\frac{5 d^{2}}{c^{4} c^{3}} (c^{4} {(d^{2})}^{4})$

Step 5.3.4

Rewrite the expression.

$\frac{5 d^{2}}{c^{3}} {(d^{2})}^{4}$

Step 5.4

Combine $\frac{5 d^{2}}{c^{3}}$ and ${(d^{2})}^{4}$ .

$\frac{5 d^{2} {(d^{2})}^{4}}{c^{3}}$

Step 6

Simplify the numerator.

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Step 6.1

Multiply the exponents in ${(d^{2})}^{4}$ .

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Step 6.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{5 d^{2} d^{2 \cdot 4}}{c^{3}}$

Step 6.1.2

Multiply $2$ by $4$ .

$\frac{5 d^{2} d^{8}}{c^{3}}$

Step 6.2

Multiply $d^{2}$ by $d^{8}$ by adding the exponents.

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Step 6.2.1

Move $d^{8}$ .

$\frac{5 (d^{8} d^{2})}{c^{3}}$

Step 6.2.2

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

$\frac{5 d^{8 + 2}}{c^{3}}$

Step 6.2.3

Add $8$ and $2$ .

$\frac{5 d^{10}}{c^{3}}$