Basic Math Examples

$- u^{2} v^{3} \cdot {(- u^{5} v^{- 5})}^{4}$

Step 1

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

$- u^{2} v^{3} \cdot {(- u^{5} \frac{1}{v^{5}})}^{4}$

Step 2

Combine $\frac{1}{v^{5}}$ and $u^{5}$ .

$- u^{2} v^{3} \cdot {(- \frac{u^{5}}{v^{5}})}^{4}$

Step 3

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

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Step 3.1

Apply the product rule to $- \frac{u^{5}}{v^{5}}$ .

$- u^{2} v^{3} \cdot ({(- 1)}^{4} {(\frac{u^{5}}{v^{5}})}^{4})$

Step 3.2

Apply the product rule to $\frac{u^{5}}{v^{5}}$ .

$- u^{2} v^{3} \cdot ({(- 1)}^{4} \frac{{(u^{5})}^{4}}{{(v^{5})}^{4}})$

Step 4

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

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Step 4.1

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

${(- 1)}^{4} \cdot - 1 u^{2} v^{3} \cdot \frac{{(u^{5})}^{4}}{{(v^{5})}^{4}}$

Step 4.2

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

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Step 4.2.1

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

${(- 1)}^{4} \cdot {(- 1)}^{1} u^{2} v^{3} \cdot \frac{{(u^{5})}^{4}}{{(v^{5})}^{4}}$

Step 4.2.2

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

${(- 1)}^{4 + 1} u^{2} v^{3} \cdot \frac{{(u^{5})}^{4}}{{(v^{5})}^{4}}$

Step 4.3

Add $4$ and $1$ .

${(- 1)}^{5} u^{2} v^{3} \cdot \frac{{(u^{5})}^{4}}{{(v^{5})}^{4}}$

Step 5

Apply basic rules of exponents.

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Step 5.1

Multiply the exponents in ${(u^{5})}^{4}$ .

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Step 5.1.1

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

${(- 1)}^{5} u^{2} v^{3} \cdot \frac{u^{5 \cdot 4}}{{(v^{5})}^{4}}$

Step 5.1.2

Multiply $5$ by $4$ .

${(- 1)}^{5} u^{2} v^{3} \cdot \frac{u^{20}}{{(v^{5})}^{4}}$

Step 5.2

Multiply the exponents in ${(v^{5})}^{4}$ .

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Step 5.2.1

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

${(- 1)}^{5} u^{2} v^{3} \cdot \frac{u^{20}}{v^{5 \cdot 4}}$

Step 5.2.2

Multiply $5$ by $4$ .

${(- 1)}^{5} u^{2} v^{3} \cdot \frac{u^{20}}{v^{20}}$

Step 6

Cancel the common factor of $v^{3}$ .

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Step 6.1

Factor $v^{3}$ out of ${(- 1)}^{5} u^{2} v^{3}$ .

$v^{3} ({(- 1)}^{5} u^{2}) \cdot \frac{u^{20}}{v^{20}}$

Step 6.2

Factor $v^{3}$ out of $v^{20}$ .

$v^{3} ({(- 1)}^{5} u^{2}) \cdot \frac{u^{20}}{v^{3} v^{17}}$

Step 6.3

Cancel the common factor.

$v^{3} ({(- 1)}^{5} u^{2}) \cdot \frac{u^{20}}{v^{3} v^{17}}$

Step 6.4

Rewrite the expression.

${(- 1)}^{5} u^{2} \cdot \frac{u^{20}}{v^{17}}$

Step 7

Combine ${(- 1)}^{5}$ and $\frac{u^{20}}{v^{17}}$ .

$u^{2} \cdot \frac{{(- 1)}^{5} u^{20}}{v^{17}}$

Step 8

Combine $u^{2}$ and $\frac{{(- 1)}^{5} u^{20}}{v^{17}}$ .

$\frac{u^{2} ({(- 1)}^{5} u^{20})}{v^{17}}$

Step 9

Multiply $u^{2}$ by $u^{20}$ by adding the exponents.

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Step 9.1

Move $u^{20}$ .

$\frac{u^{20} u^{2} {(- 1)}^{5}}{v^{17}}$

Step 9.2

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

$\frac{u^{20 + 2} {(- 1)}^{5}}{v^{17}}$

Step 9.3

Add $20$ and $2$ .

$\frac{u^{22} {(- 1)}^{5}}{v^{17}}$

Step 10

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

$\frac{u^{22} \cdot - 1}{v^{17}}$

Step 11

Move $- 1$ to the left of $u^{22}$ .

$\frac{- 1 \cdot u^{22}}{v^{17}}$

Step 12

Move the negative in front of the fraction.

$- \frac{u^{22}}{v^{17}}$