Simplify (5y^4n^-4)/((3n^-5)^-3)

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

Step 1

Simplify.

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

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

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

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

Step 2

Simplify the numerator.

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

$\frac{5 y^{4} \cdot 3^{3} {(n^{- 5})}^{3}}{n^{4}}$

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

$\frac{5 y^{4} \cdot 27 {(n^{- 5})}^{3}}{n^{4}}$

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

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

$\frac{5 y^{4} \cdot 27 n^{- 5 \cdot 3}}{n^{4}}$

Multiply $- 5$ by $3$ .

$\frac{5 y^{4} \cdot 27 n^{- 15}}{n^{4}}$

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

$\frac{5 y^{4} \cdot 27 \frac{1}{n^{15}}}{n^{4}}$

Combine exponents.

Multiply $27$ by $5$ .

$\frac{135 y^{4} \frac{1}{n^{15}}}{n^{4}}$

Combine $135$ and $\frac{1}{n^{15}}$ .

$\frac{y^{4} \frac{135}{n^{15}}}{n^{4}}$

Combine $y^{4}$ and $\frac{135}{n^{15}}$ .

$\frac{\frac{y^{4} \cdot 135}{n^{15}}}{n^{4}}$

Move $135$ to the left of $y^{4}$ .

$\frac{\frac{135 y^{4}}{n^{15}}}{n^{4}}$

Step 3

Multiply the numerator by the reciprocal of the denominator.

$\frac{135 y^{4}}{n^{15}} \cdot \frac{1}{n^{4}}$

Step 4

Combine.

$\frac{135 y^{4} \cdot 1}{n^{15} n^{4}}$

Step 5

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

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

$\frac{135 y^{4} \cdot 1}{n^{15 + 4}}$

Add $15$ and $4$ .

$\frac{135 y^{4} \cdot 1}{n^{19}}$

Step 6

Multiply $135 y^{4}$ by $1$ .

$\frac{135 y^{4}}{n^{19}}$