Simplify ((3d^-4n^5)^-3)/(12d^3n^4)

$\frac{{(3 d^{- 4} n^{5})}^{- 3}}{12 d^{3} n^{4}}$

Step 1

Move ${(3 d^{- 4} n^{5})}^{- 3}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{12 d^{3} n^{4} {(3 d^{- 4} n^{5})}^{3}}$

Step 2

Simplify the denominator.

Apply the product rule to $3 d^{- 4} n^{5}$ .

$\frac{1}{12 d^{3} n^{4} {(3 d^{- 4})}^{3} {(n^{5})}^{3}}$

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

$\frac{1}{12 d^{3} n^{4} {(3 \frac{1}{d^{4}})}^{3} {(n^{5})}^{3}}$

Combine $3$ and $\frac{1}{d^{4}}$ .

$\frac{1}{12 d^{3} n^{4} {(\frac{3}{d^{4}})}^{3} {(n^{5})}^{3}}$

Apply the product rule to $\frac{3}{d^{4}}$ .

$\frac{1}{12 d^{3} n^{4} \frac{3^{3}}{{(d^{4})}^{3}} {(n^{5})}^{3}}$

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

$\frac{1}{12 d^{3} n^{4} \frac{27}{{(d^{4})}^{3}} {(n^{5})}^{3}}$

Multiply the exponents in ${(d^{4})}^{3}$ .

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

$\frac{1}{12 d^{3} n^{4} \frac{27}{d^{4 \cdot 3}} {(n^{5})}^{3}}$

Multiply $4$ by $3$ .

$\frac{1}{12 d^{3} n^{4} \frac{27}{d^{12}} {(n^{5})}^{3}}$

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

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

$\frac{1}{12 d^{3} n^{4} \frac{27}{d^{12}} n^{5 \cdot 3}}$

Multiply $5$ by $3$ .

$\frac{1}{12 d^{3} n^{4} \frac{27}{d^{12}} n^{15}}$

Combine exponents.

Combine $12$ and $\frac{27}{d^{12}}$ .

$\frac{1}{d^{3} n^{4} \frac{12 \cdot 27}{d^{12}} n^{15}}$

Multiply $12$ by $27$ .

$\frac{1}{d^{3} n^{4} \frac{324}{d^{12}} n^{15}}$

Combine $d^{3}$ and $\frac{324}{d^{12}}$ .

$\frac{1}{n^{4} \frac{d^{3} \cdot 324}{d^{12}} n^{15}}$

Combine $n^{4}$ and $\frac{d^{3} \cdot 324}{d^{12}}$ .

$\frac{1}{\frac{n^{4} (d^{3} \cdot 324)}{d^{12}} n^{15}}$

Combine $\frac{n^{4} (d^{3} \cdot 324)}{d^{12}}$ and $n^{15}$ .

$\frac{1}{\frac{n^{4} (d^{3} \cdot 324) n^{15}}{d^{12}}}$

Multiply $n^{4}$ by $n^{15}$ by adding the exponents.

Move $n^{15}$ .

$\frac{1}{\frac{n^{15} n^{4} (d^{3} \cdot 324)}{d^{12}}}$

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

$\frac{1}{\frac{n^{15 + 4} (d^{3} \cdot 324)}{d^{12}}}$

Add $15$ and $4$ .

$\frac{1}{\frac{n^{19} (d^{3} \cdot 324)}{d^{12}}}$

Rewrite.

$\frac{1}{\frac{n^{4} (d^{3} \cdot 324) n^{15}}{d^{12}}}$

Multiply $n^{4}$ by $n^{15}$ by adding the exponents.

Move $n^{15}$ .

$\frac{1}{\frac{n^{15} n^{4} (d^{3} \cdot 324)}{d^{12}}}$

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

$\frac{1}{\frac{n^{15 + 4} (d^{3} \cdot 324)}{d^{12}}}$

Add $15$ and $4$ .

$\frac{1}{\frac{n^{19} (d^{3} \cdot 324)}{d^{12}}}$

Remove unnecessary parentheses.

$\frac{1}{\frac{n^{19} d^{3} \cdot 324}{d^{12}}}$

Reduce the expression $\frac{n^{19} d^{3} \cdot 324}{d^{12}}$ by cancelling the common factors.

Factor $d^{3}$ out of $n^{19} d^{3} \cdot 324$ .

$\frac{1}{\frac{d^{3} (n^{19} \cdot 324)}{d^{12}}}$

Factor $d^{3}$ out of $d^{12}$ .

$\frac{1}{\frac{d^{3} (n^{19} \cdot 324)}{d^{3} d^{9}}}$

Cancel the common factor.

$\frac{1}{\frac{d^{3} (n^{19} \cdot 324)}{d^{3} d^{9}}}$

Rewrite the expression.

$\frac{1}{\frac{n^{19} \cdot 324}{d^{9}}}$

Move $324$ to the left of $n^{19}$ .

$\frac{1}{\frac{324 n^{19}}{d^{9}}}$

Step 3

Multiply the numerator by the reciprocal of the denominator.

$1 \frac{d^{9}}{324 n^{19}}$

Step 4

Multiply $\frac{d^{9}}{324 n^{19}}$ by $1$ .

$\frac{d^{9}}{324 n^{19}}$

Simplify ((3d^-4n^5)^-3)/(12d^3n^4)

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