Basic Math Examples

${(- u^{2} v^{- 1})}^{4} \cdot (- u v^{4})$

Step 1

Rewrite using the commutative property of multiplication.

$- {(- u^{2} v^{- 1})}^{4} u v^{4}$

Step 2

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

$- {(- u^{2} \frac{1}{v})}^{4} u v^{4}$

Step 3

Combine $\frac{1}{v}$ and $u^{2}$ .

$- {(- \frac{u^{2}}{v})}^{4} u v^{4}$

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 $- \frac{u^{2}}{v}$ .

$- ({(- 1)}^{4} {(\frac{u^{2}}{v})}^{4}) u v^{4}$

Step 4.2

Apply the product rule to $\frac{u^{2}}{v}$ .

$- ({(- 1)}^{4} \frac{{(u^{2})}^{4}}{v^{4}}) u v^{4}$

Step 5

Multiply $- 1$ by ${(- 1)}^{4}$ by adding the exponents.

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Step 5.1

Move ${(- 1)}^{4}$ .

${(- 1)}^{4} \cdot - 1 \frac{{(u^{2})}^{4}}{v^{4}} u v^{4}$

Step 5.2

Multiply ${(- 1)}^{4}$ by $- 1$ .

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Step 5.2.1

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

${(- 1)}^{4} \cdot {(- 1)}^{1} \frac{{(u^{2})}^{4}}{v^{4}} u v^{4}$

Step 5.2.2

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

${(- 1)}^{4 + 1} \frac{{(u^{2})}^{4}}{v^{4}} u v^{4}$

Step 5.3

Add $4$ and $1$ .

${(- 1)}^{5} \frac{{(u^{2})}^{4}}{v^{4}} u v^{4}$

Step 6

Raise $- 1$ to the power of $5$ .

$- \frac{{(u^{2})}^{4}}{v^{4}} u v^{4}$

Step 7

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

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Step 7.1

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

$- \frac{u^{2 \cdot 4}}{v^{4}} u v^{4}$

Step 7.2

Multiply $2$ by $4$ .

$- \frac{u^{8}}{v^{4}} u v^{4}$

Step 8

Multiply $- \frac{u^{8}}{v^{4}} u$ .

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Step 8.1

Combine $u$ and $\frac{u^{8}}{v^{4}}$ .

$- \frac{u \cdot u^{8}}{v^{4}} v^{4}$

Step 8.2

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

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Step 8.2.1

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

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Step 8.2.1.1

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

$- \frac{u^{1} u^{8}}{v^{4}} v^{4}$

Step 8.2.1.2

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

$- \frac{u^{1 + 8}}{v^{4}} v^{4}$

Step 8.2.2

Add $1$ and $8$ .

$- \frac{u^{9}}{v^{4}} v^{4}$

Step 9

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

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Step 9.1

Move the leading negative in $- \frac{u^{9}}{v^{4}}$ into the numerator.

$\frac{- u^{9}}{v^{4}} v^{4}$

Step 9.2

Cancel the common factor.

$\frac{- u^{9}}{v^{4}} v^{4}$

Step 9.3

Rewrite the expression.

$- u^{9}$