Simplify ((-3u^-3h^3)^5)/(12u^-9h^10)

$\frac{{(- 3 u^{- 3} h^{3})}^{5}}{12 u^{- 9} h^{10}}$

Step 1

Move $u^{- 9}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$\frac{{(- 3 u^{- 3} h^{3})}^{5} u^{9}}{12 h^{10}}$

Step 2

Simplify the numerator.

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

$\frac{{(- 3 u^{- 3})}^{5} {(h^{3})}^{5} u^{9}}{12 h^{10}}$

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

$\frac{{(- 3 \frac{1}{u^{3}})}^{5} {(h^{3})}^{5} u^{9}}{12 h^{10}}$

Combine $- 3$ and $\frac{1}{u^{3}}$ .

$\frac{{(\frac{- 3}{u^{3}})}^{5} {(h^{3})}^{5} u^{9}}{12 h^{10}}$

Move the negative in front of the fraction.

$\frac{{(- \frac{3}{u^{3}})}^{5} {(h^{3})}^{5} u^{9}}{12 h^{10}}$

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

Apply the product rule to $- \frac{3}{u^{3}}$ .

$\frac{{(- 1)}^{5} {(\frac{3}{u^{3}})}^{5} {(h^{3})}^{5} u^{9}}{12 h^{10}}$

Apply the product rule to $\frac{3}{u^{3}}$ .

$\frac{{(- 1)}^{5} \frac{3^{5}}{{(u^{3})}^{5}} {(h^{3})}^{5} u^{9}}{12 h^{10}}$

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

$\frac{- \frac{3^{5}}{{(u^{3})}^{5}} {(h^{3})}^{5} u^{9}}{12 h^{10}}$

Raise $3$ to the power of $5$ .

$\frac{- \frac{243}{{(u^{3})}^{5}} {(h^{3})}^{5} u^{9}}{12 h^{10}}$

Multiply the exponents in ${(u^{3})}^{5}$ .

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

$\frac{- \frac{243}{u^{3 \cdot 5}} {(h^{3})}^{5} u^{9}}{12 h^{10}}$

Multiply $3$ by $5$ .

$\frac{- \frac{243}{u^{15}} {(h^{3})}^{5} u^{9}}{12 h^{10}}$

Multiply the exponents in ${(h^{3})}^{5}$ .

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

$\frac{- \frac{243}{u^{15}} h^{3 \cdot 5} u^{9}}{12 h^{10}}$

Multiply $3$ by $5$ .

$\frac{- \frac{243}{u^{15}} h^{15} u^{9}}{12 h^{10}}$

Combine exponents.

Combine $h^{15}$ and $\frac{243}{u^{15}}$ .

$\frac{- \frac{h^{15} \cdot 243}{u^{15}} u^{9}}{12 h^{10}}$

Combine $u^{9}$ and $\frac{h^{15} \cdot 243}{u^{15}}$ .

$\frac{- \frac{u^{9} (h^{15} \cdot 243)}{u^{15}}}{12 h^{10}}$

Remove unnecessary parentheses.

$\frac{- \frac{u^{9} h^{15} \cdot 243}{u^{15}}}{12 h^{10}}$

Reduce the expression $\frac{u^{9} h^{15} \cdot 243}{u^{15}}$ by cancelling the common factors.

Factor $u^{9}$ out of $u^{9} h^{15} \cdot 243$ .

$\frac{- \frac{u^{9} (h^{15} \cdot 243)}{u^{15}}}{12 h^{10}}$

Factor $u^{9}$ out of $u^{15}$ .

$\frac{- \frac{u^{9} (h^{15} \cdot 243)}{u^{9} u^{6}}}{12 h^{10}}$

Cancel the common factor.

$\frac{- \frac{u^{9} (h^{15} \cdot 243)}{u^{9} u^{6}}}{12 h^{10}}$

Rewrite the expression.

$\frac{- \frac{h^{15} \cdot 243}{u^{6}}}{12 h^{10}}$

Move $243$ to the left of $h^{15}$ .

$\frac{- \frac{243 h^{15}}{u^{6}}}{12 h^{10}}$

Step 3

Multiply the numerator by the reciprocal of the denominator.

$- \frac{243 h^{15}}{u^{6}} \cdot \frac{1}{12 h^{10}}$

Step 4

Cancel the common factor of $3 h^{10}$ .

Move the leading negative in $- \frac{243 h^{15}}{u^{6}}$ into the numerator.

$\frac{- 243 h^{15}}{u^{6}} \cdot \frac{1}{12 h^{10}}$

Factor $3 h^{10}$ out of $- 243 h^{15}$ .

$\frac{3 h^{10} (- 81 h^{5})}{u^{6}} \cdot \frac{1}{12 h^{10}}$

Factor $3 h^{10}$ out of $12 h^{10}$ .

$\frac{3 h^{10} (- 81 h^{5})}{u^{6}} \cdot \frac{1}{3 h^{10} (4)}$

Cancel the common factor.

$\frac{3 h^{10} (- 81 h^{5})}{u^{6}} \cdot \frac{1}{3 h^{10} \cdot 4}$

Rewrite the expression.

$\frac{- 81 h^{5}}{u^{6}} \cdot \frac{1}{4}$

Step 5

Multiply $\frac{- 81 h^{5}}{u^{6}}$ by $\frac{1}{4}$ .

$\frac{- 81 h^{5}}{u^{6} \cdot 4}$

Step 6

Simplify the expression.

Move $4$ to the left of $u^{6}$ .

$\frac{- 81 h^{5}}{4 \cdot u^{6}}$

Move the negative in front of the fraction.

$- \frac{81 h^{5}}{4 u^{6}}$

Simplify ((-3u^-3h^3)^5)/(12u^-9h^10)

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