Simplify (15r^6c^-8)/((3r^-3c)^-5)

$\frac{15 r^{6} c^{- 8}}{{(3 r^{- 3} c)}^{- 5}}$

Step 1

Simplify.

Move $c^{- 8}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{15 r^{6}}{{(3 r^{- 3} c)}^{- 5} c^{8}}$

Move ${(3 r^{- 3} c)}^{- 5}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$\frac{15 r^{6} {(3 r^{- 3} c)}^{5}}{c^{8}}$

Step 2

Simplify the numerator.

Apply the product rule to $3 r^{- 3} c$ .

$\frac{15 r^{6} {(3 r^{- 3})}^{5} c^{5}}{c^{8}}$

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

$\frac{15 r^{6} {(3 \frac{1}{r^{3}})}^{5} c^{5}}{c^{8}}$

Combine $3$ and $\frac{1}{r^{3}}$ .

$\frac{15 r^{6} {(\frac{3}{r^{3}})}^{5} c^{5}}{c^{8}}$

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

$\frac{15 r^{6} \frac{3^{5}}{{(r^{3})}^{5}} c^{5}}{c^{8}}$

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

$\frac{15 r^{6} \frac{243}{{(r^{3})}^{5}} c^{5}}{c^{8}}$

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

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

$\frac{15 r^{6} \frac{243}{r^{3 \cdot 5}} c^{5}}{c^{8}}$

Multiply $3$ by $5$ .

$\frac{15 r^{6} \frac{243}{r^{15}} c^{5}}{c^{8}}$

Combine exponents.

Combine $15$ and $\frac{243}{r^{15}}$ .

$\frac{r^{6} \frac{15 \cdot 243}{r^{15}} c^{5}}{c^{8}}$

Multiply $15$ by $243$ .

$\frac{r^{6} \frac{3645}{r^{15}} c^{5}}{c^{8}}$

Combine $r^{6}$ and $\frac{3645}{r^{15}}$ .

$\frac{\frac{r^{6} \cdot 3645}{r^{15}} c^{5}}{c^{8}}$

Combine $\frac{r^{6} \cdot 3645}{r^{15}}$ and $c^{5}$ .

$\frac{\frac{r^{6} \cdot 3645 c^{5}}{r^{15}}}{c^{8}}$

Reduce the expression $\frac{r^{6} \cdot 3645 c^{5}}{r^{15}}$ by cancelling the common factors.

Factor $r^{6}$ out of $r^{6} \cdot 3645 c^{5}$ .

$\frac{\frac{r^{6} (3645 c^{5})}{r^{15}}}{c^{8}}$

Factor $r^{6}$ out of $r^{15}$ .

$\frac{\frac{r^{6} (3645 c^{5})}{r^{6} r^{9}}}{c^{8}}$

Cancel the common factor.

$\frac{\frac{r^{6} (3645 c^{5})}{r^{6} r^{9}}}{c^{8}}$

Rewrite the expression.

$\frac{\frac{3645 c^{5}}{r^{9}}}{c^{8}}$

Step 3

Multiply the numerator by the reciprocal of the denominator.

$\frac{3645 c^{5}}{r^{9}} \cdot \frac{1}{c^{8}}$

Step 4

Combine.

$\frac{3645 c^{5} \cdot 1}{r^{9} c^{8}}$

Step 5

Cancel the common factor of $c^{5}$ and $c^{8}$ .

Factor $c^{5}$ out of $3645 c^{5} \cdot 1$ .

$\frac{c^{5} (3645 \cdot 1)}{r^{9} c^{8}}$

Cancel the common factors.

Factor $c^{5}$ out of $r^{9} c^{8}$ .

$\frac{c^{5} (3645 \cdot 1)}{c^{5} (r^{9} c^{3})}$

Cancel the common factor.

$\frac{c^{5} (3645 \cdot 1)}{c^{5} (r^{9} c^{3})}$

Rewrite the expression.

$\frac{3645 \cdot 1}{r^{9} c^{3}}$

Step 6

Multiply $3645$ by $1$ .

$\frac{3645}{r^{9} c^{3}}$

Simplify (15r^6c^-8)/((3r^-3c)^-5)

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