Simplify (2x^3y^4)/((2yx^-1)^4)

$\frac{2 x^{3} y^{4}}{{(2 y x^{- 1})}^{4}}$

Step 1

Simplify the denominator.

Apply the product rule to $2 y x^{- 1}$ .

$\frac{2 x^{3} y^{4}}{{(2 y)}^{4} {(x^{- 1})}^{4}}$

Apply the product rule to $2 y$ .

$\frac{2 x^{3} y^{4}}{2^{4} y^{4} {(x^{- 1})}^{4}}$

Raise $2$ to the power of $4$ .

$\frac{2 x^{3} y^{4}}{16 y^{4} {(x^{- 1})}^{4}}$

Multiply the exponents in ${(x^{- 1})}^{4}$ .

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

$\frac{2 x^{3} y^{4}}{16 y^{4} x^{- 1 \cdot 4}}$

Multiply $- 1$ by $4$ .

$\frac{2 x^{3} y^{4}}{16 y^{4} x^{- 4}}$

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

$\frac{2 x^{3} y^{4}}{16 y^{4} \frac{1}{x^{4}}}$

Combine exponents.

Combine $16$ and $\frac{1}{x^{4}}$ .

$\frac{2 x^{3} y^{4}}{y^{4} \frac{16}{x^{4}}}$

Combine $y^{4}$ and $\frac{16}{x^{4}}$ .

$\frac{2 x^{3} y^{4}}{\frac{y^{4} \cdot 16}{x^{4}}}$

Move $16$ to the left of $y^{4}$ .

$\frac{2 x^{3} y^{4}}{\frac{16 y^{4}}{x^{4}}}$

Step 2

Multiply the numerator by the reciprocal of the denominator.

$2 x^{3} y^{4} \frac{x^{4}}{16 y^{4}}$

Step 3

Cancel the common factor of $2 y^{4}$ .

Factor $2 y^{4}$ out of $2 x^{3} y^{4}$ .

$2 y^{4} (x^{3}) \frac{x^{4}}{16 y^{4}}$

Factor $2 y^{4}$ out of $16 y^{4}$ .

$2 y^{4} (x^{3}) \frac{x^{4}}{2 y^{4} (8)}$

Cancel the common factor.

$2 y^{4} x^{3} \frac{x^{4}}{2 y^{4} \cdot 8}$

Rewrite the expression.

$x^{3} \frac{x^{4}}{8}$

Step 4

Combine $x^{3}$ and $\frac{x^{4}}{8}$ .

$\frac{x^{3} x^{4}}{8}$

Step 5

Multiply $x^{3}$ by $x^{4}$ by adding the exponents.

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

$\frac{x^{3 + 4}}{8}$

Add $3$ and $4$ .

$\frac{x^{7}}{8}$