Simplify ((4x^-3y^2)^2)/(2x^-5y^3)

$\frac{{(4 x^{- 3} y^{2})}^{2}}{2 x^{- 5} y^{3}}$

Step 1

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

$\frac{{(4 x^{- 3} y^{2})}^{2} x^{5}}{2 y^{3}}$

Step 2

Simplify the numerator.

Apply the product rule to $4 x^{- 3} y^{2}$ .

$\frac{{(4 x^{- 3})}^{2} {(y^{2})}^{2} x^{5}}{2 y^{3}}$

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

$\frac{{(4 \frac{1}{x^{3}})}^{2} {(y^{2})}^{2} x^{5}}{2 y^{3}}$

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

$\frac{{(\frac{4}{x^{3}})}^{2} {(y^{2})}^{2} x^{5}}{2 y^{3}}$

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

$\frac{\frac{4^{2}}{{(x^{3})}^{2}} {(y^{2})}^{2} x^{5}}{2 y^{3}}$

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

$\frac{\frac{16}{{(x^{3})}^{2}} {(y^{2})}^{2} x^{5}}{2 y^{3}}$

Multiply the exponents in ${(x^{3})}^{2}$ .

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

$\frac{\frac{16}{x^{3 \cdot 2}} {(y^{2})}^{2} x^{5}}{2 y^{3}}$

Multiply $3$ by $2$ .

$\frac{\frac{16}{x^{6}} {(y^{2})}^{2} x^{5}}{2 y^{3}}$

Multiply the exponents in ${(y^{2})}^{2}$ .

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

$\frac{\frac{16}{x^{6}} y^{2 \cdot 2} x^{5}}{2 y^{3}}$

Multiply $2$ by $2$ .

$\frac{\frac{16}{x^{6}} y^{4} x^{5}}{2 y^{3}}$

Combine exponents.

Combine $\frac{16}{x^{6}}$ and $y^{4}$ .

$\frac{\frac{16 y^{4}}{x^{6}} x^{5}}{2 y^{3}}$

Combine $\frac{16 y^{4}}{x^{6}}$ and $x^{5}$ .

$\frac{\frac{16 y^{4} x^{5}}{x^{6}}}{2 y^{3}}$

Reduce the expression $\frac{16 y^{4} x^{5}}{x^{6}}$ by cancelling the common factors.

Factor $x^{5}$ out of $16 y^{4} x^{5}$ .

$\frac{\frac{x^{5} (16 y^{4})}{x^{6}}}{2 y^{3}}$

Factor $x^{5}$ out of $x^{6}$ .

$\frac{\frac{x^{5} (16 y^{4})}{x^{5} x}}{2 y^{3}}$

Cancel the common factor.

$\frac{\frac{x^{5} (16 y^{4})}{x^{5} x}}{2 y^{3}}$

Rewrite the expression.

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

Step 3

Multiply the numerator by the reciprocal of the denominator.

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

Step 4

Combine.

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

Step 5

Cancel the common factor of $16$ and $2$ .

Factor $2$ out of $16 y^{4} \cdot 1$ .

$\frac{2 (8 y^{4} \cdot 1)}{x (2 y^{3})}$

Cancel the common factors.

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

$\frac{2 (8 y^{4} \cdot 1)}{2 (x (y^{3}))}$

Cancel the common factor.

$\frac{2 (8 y^{4} \cdot 1)}{2 (x (y^{3}))}$

Rewrite the expression.

$\frac{8 y^{4} \cdot 1}{x (y^{3})}$

Step 6

Cancel the common factor of $y^{4}$ and $y^{3}$ .

Factor $y^{3}$ out of $8 y^{4} \cdot 1$ .

$\frac{y^{3} (8 y \cdot 1)}{x (y^{3})}$

Cancel the common factors.

Factor $y^{3}$ out of $x (y^{3})$ .

$\frac{y^{3} (8 y \cdot 1)}{y^{3} x}$

Cancel the common factor.

$\frac{y^{3} (8 y \cdot 1)}{y^{3} x}$

Rewrite the expression.

$\frac{8 y \cdot 1}{x}$

Step 7

Multiply $8$ by $1$ .

$\frac{8 y}{x}$

Simplify ((4x^-3y^2)^2)/(2x^-5y^3)

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