Finite Math Examples

$(\frac{u^{- 3}}{z^{2}}) {(- 2 x^{3} u z^{2})}^{- 3}$

Step 1

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

$\frac{\frac{1}{u^{3}}}{z^{2}} {(- 2 x^{3} u z^{2})}^{- 3}$

Step 2

Multiply the numerator by the reciprocal of the denominator.

$\frac{1}{u^{3}} \cdot \frac{1}{z^{2}} {(- 2 x^{3} u z^{2})}^{- 3}$

Step 3

Combine.

$\frac{1 \cdot 1}{u^{3} z^{2}} {(- 2 x^{3} u z^{2})}^{- 3}$

Step 4

Multiply $1$ by $1$ .

$\frac{1}{u^{3} z^{2}} {(- 2 x^{3} u z^{2})}^{- 3}$

Step 5

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

$\frac{1}{u^{3} z^{2}} \cdot \frac{1}{{(- 2 x^{3} u z^{2})}^{3}}$

Step 6

Combine fractions.

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Step 6.1

Combine.

$\frac{1 \cdot 1}{u^{3} z^{2} {(- 2 x^{3} u z^{2})}^{3}}$

Step 6.2

Multiply $1$ by $1$ .

$\frac{1}{u^{3} z^{2} {(- 2 x^{3} u z^{2})}^{3}}$

Step 7

Simplify the denominator.

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Step 7.1

Apply the product rule to $- 2 x^{3} u z^{2}$ .

$\frac{1}{u^{3} z^{2} {(- 2 x^{3} u)}^{3} {(z^{2})}^{3}}$

Step 7.2

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

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Step 7.2.1

Apply the product rule to $- 2 x^{3} u$ .

$\frac{1}{u^{3} z^{2} ({(- 2 x^{3})}^{3} u^{3}) {(z^{2})}^{3}}$

Step 7.2.2

Apply the product rule to $- 2 x^{3}$ .

$\frac{1}{u^{3} z^{2} ({(- 2)}^{3} {(x^{3})}^{3} u^{3}) {(z^{2})}^{3}}$

Step 7.3

Raise $- 2$ to the power of $3$ .

$\frac{1}{u^{3} z^{2} (- 8 {(x^{3})}^{3} u^{3}) {(z^{2})}^{3}}$

Step 7.4

Multiply the exponents in ${(x^{3})}^{3}$ .

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Step 7.4.1

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

$\frac{1}{u^{3} z^{2} (- 8 x^{3 \cdot 3} u^{3}) {(z^{2})}^{3}}$

Step 7.4.2

Multiply $3$ by $3$ .

$\frac{1}{u^{3} z^{2} (- 8 x^{9} u^{3}) {(z^{2})}^{3}}$

Step 7.5

Multiply the exponents in ${(z^{2})}^{3}$ .

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Step 7.5.1

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

$\frac{1}{u^{3} z^{2} (- 8 x^{9} u^{3}) z^{2 \cdot 3}}$

Step 7.5.2

Multiply $2$ by $3$ .

$\frac{1}{u^{3} z^{2} (- 8 x^{9} u^{3}) z^{6}}$

$\frac{1}{u^{3} z^{2} \cdot - 8 x^{9} u^{3} z^{6}}$

Step 7.6

Combine exponents.

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Step 7.6.1

Multiply $u^{3}$ by $u^{3}$ by adding the exponents.

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Step 7.6.1.1

Move $u^{3}$ .

$\frac{1}{u^{3} u^{3} z^{2} \cdot - 8 x^{9} z^{6}}$

Step 7.6.1.2

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

$\frac{1}{u^{3 + 3} z^{2} \cdot - 8 x^{9} z^{6}}$

Step 7.6.1.3

Add $3$ and $3$ .

$\frac{1}{u^{6} z^{2} \cdot - 8 x^{9} z^{6}}$

Step 7.6.2

Multiply $z^{2}$ by $z^{6}$ by adding the exponents.

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Step 7.6.2.1

Move $z^{6}$ .

$\frac{1}{u^{6} (z^{6} z^{2}) \cdot - 8 x^{9}}$

Step 7.6.2.2

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

$\frac{1}{u^{6} z^{6 + 2} \cdot - 8 x^{9}}$

Step 7.6.2.3

Add $6$ and $2$ .

$\frac{1}{u^{6} z^{8} \cdot - 8 x^{9}}$

Step 8

Simplify the expression.

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Step 8.1

Move $- 8$ to the left of $u^{6} z^{8}$ .

$\frac{1}{- 8 \cdot (u^{6} z^{8}) x^{9}}$

Step 8.2

Move the negative in front of the fraction.

$- \frac{1}{8 u^{6} z^{8} x^{9}}$