Evaluate ((5x^-4y^3)/(2x^2y^0))^2*((4xy)/(y^3))^2

${(\frac{5 x^{- 4} y^{3}}{2 x^{2} y^{0}})}^{2} \cdot {(\frac{4 x y}{y^{3}})}^{2}$

Step 1

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

${(\frac{5 y^{3}}{2 x^{2} y^{0} x^{4}})}^{2} \cdot {(\frac{4 x y}{y^{3}})}^{2}$

Step 2

Simplify the denominator.

Multiply $x^{2}$ by $x^{4}$ by adding the exponents.

Move $x^{4}$ .

${(\frac{5 y^{3}}{2 (x^{4} x^{2}) y^{0}})}^{2} \cdot {(\frac{4 x y}{y^{3}})}^{2}$

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

${(\frac{5 y^{3}}{2 x^{4 + 2} y^{0}})}^{2} \cdot {(\frac{4 x y}{y^{3}})}^{2}$

Add $4$ and $2$ .

${(\frac{5 y^{3}}{2 x^{6} y^{0}})}^{2} \cdot {(\frac{4 x y}{y^{3}})}^{2}$

Simplify $2 x^{6} y^{0}$ .

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

Step 3

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $\frac{5 y^{3}}{2 x^{6}}$ .

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

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

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

Apply the product rule to $2 x^{6}$ .

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

Step 4

Simplify the numerator.

Raise $5$ to the power of $2$ .

$\frac{25 {(y^{3})}^{2}}{2^{2} {(x^{6})}^{2}} \cdot {(\frac{4 x y}{y^{3}})}^{2}$

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

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

$\frac{25 y^{3 \cdot 2}}{2^{2} {(x^{6})}^{2}} \cdot {(\frac{4 x y}{y^{3}})}^{2}$

Multiply $3$ by $2$ .

$\frac{25 y^{6}}{2^{2} {(x^{6})}^{2}} \cdot {(\frac{4 x y}{y^{3}})}^{2}$

Step 5

Simplify the denominator.

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

$\frac{25 y^{6}}{4 {(x^{6})}^{2}} \cdot {(\frac{4 x y}{y^{3}})}^{2}$

Multiply the exponents in ${(x^{6})}^{2}$ .

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

$\frac{25 y^{6}}{4 x^{6 \cdot 2}} \cdot {(\frac{4 x y}{y^{3}})}^{2}$

Multiply $6$ by $2$ .

$\frac{25 y^{6}}{4 x^{12}} \cdot {(\frac{4 x y}{y^{3}})}^{2}$

Step 6

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

Factor $y$ out of $4 x y$ .

$\frac{25 y^{6}}{4 x^{12}} \cdot {(\frac{y (4 x)}{y^{3}})}^{2}$

Cancel the common factors.

Factor $y$ out of $y^{3}$ .

$\frac{25 y^{6}}{4 x^{12}} \cdot {(\frac{y (4 x)}{y \cdot y^{2}})}^{2}$

Cancel the common factor.

$\frac{25 y^{6}}{4 x^{12}} \cdot {(\frac{y (4 x)}{y \cdot y^{2}})}^{2}$

Rewrite the expression.

$\frac{25 y^{6}}{4 x^{12}} \cdot {(\frac{4 x}{y^{2}})}^{2}$

Step 7

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $\frac{4 x}{y^{2}}$ .

$\frac{25 y^{6}}{4 x^{12}} \cdot \frac{{(4 x)}^{2}}{{(y^{2})}^{2}}$

Apply the product rule to $4 x$ .

$\frac{25 y^{6}}{4 x^{12}} \cdot \frac{4^{2} x^{2}}{{(y^{2})}^{2}}$

Step 8

Combine.

$\frac{25 y^{6} (4^{2} x^{2})}{4 x^{12} {(y^{2})}^{2}}$

Step 9

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

Factor $4$ out of $25 y^{6} (4^{2} x^{2})$ .

$\frac{4 (25 y^{6} (4 x^{2}))}{4 x^{12} {(y^{2})}^{2}}$

Cancel the common factors.

Factor $4$ out of $4 x^{12} {(y^{2})}^{2}$ .

$\frac{4 (25 y^{6} (4 x^{2}))}{4 (x^{12} {(y^{2})}^{2})}$

Cancel the common factor.

$\frac{4 (25 y^{6} (4 x^{2}))}{4 (x^{12} {(y^{2})}^{2})}$

Rewrite the expression.

$\frac{25 y^{6} (4 x^{2})}{x^{12} {(y^{2})}^{2}}$

Step 10

Cancel the common factor of $x^{2}$ and $x^{12}$ .

Factor $x^{2}$ out of $25 y^{6} (4 x^{2})$ .

$\frac{x^{2} (25 y^{6} \cdot (4))}{x^{12} {(y^{2})}^{2}}$

Cancel the common factors.

Factor $x^{2}$ out of $x^{12} {(y^{2})}^{2}$ .

$\frac{x^{2} (25 y^{6} \cdot (4))}{x^{2} (x^{10} {(y^{2})}^{2})}$

Cancel the common factor.

$\frac{x^{2} (25 y^{6} \cdot (4))}{x^{2} (x^{10} {(y^{2})}^{2})}$

Rewrite the expression.

$\frac{25 y^{6} \cdot (4)}{x^{10} {(y^{2})}^{2}}$

Step 11

Simplify the expression.

Multiply $4$ by $25$ .

$\frac{100 y^{6}}{x^{10} {(y^{2})}^{2}}$

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

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

$\frac{100 y^{6}}{x^{10} y^{2 \cdot 2}}$

Multiply $2$ by $2$ .

$\frac{100 y^{6}}{x^{10} y^{4}}$

Step 12

Cancel the common factor of $y^{6}$ and $y^{4}$ .

Factor $y^{4}$ out of $100 y^{6}$ .

$\frac{y^{4} (100 y^{2})}{x^{10} y^{4}}$

Cancel the common factors.

Factor $y^{4}$ out of $x^{10} y^{4}$ .

$\frac{y^{4} (100 y^{2})}{y^{4} x^{10}}$

Cancel the common factor.

$\frac{y^{4} (100 y^{2})}{y^{4} x^{10}}$

Rewrite the expression.

$\frac{100 y^{2}}{x^{10}}$

Evaluate ((5x^-4y^3)/(2x^2y^0))^2*((4xy)/(y^3))^2

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