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

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

Step 1

Simplify the numerator.

Apply the product rule to $4 x^{2} y^{5}$ .

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

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

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

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

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

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

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

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

Multiply $2$ by $2$ .

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

Multiply the exponents in ${(y^{5})}^{2}$ .

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

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

Multiply $5$ by $2$ .

$\frac{16 x^{4} y^{10}}{{(2 x y^{2})}^{3} {(3 x^{3})}^{2}}$

Step 2

Simplify the denominator.

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

$\frac{16 x^{4} y^{10}}{{(2 x)}^{3} {(y^{2})}^{3} {(3 x^{3})}^{2}}$

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

$\frac{16 x^{4} y^{10}}{{(2 x)}^{3} {(y^{2})}^{3} \cdot 3^{2} {(x^{3})}^{2}}$

Apply the product rule to $2 x$ .

$\frac{16 x^{4} y^{10}}{2^{3} x^{3} {(y^{2})}^{3} \cdot 3^{2} {(x^{3})}^{2}}$

Raise $2$ to the power of $3$ .

$\frac{16 x^{4} y^{10}}{8 x^{3} {(y^{2})}^{3} \cdot 3^{2} {(x^{3})}^{2}}$

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

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

$\frac{16 x^{4} y^{10}}{8 x^{3} y^{2 \cdot 3} \cdot 3^{2} {(x^{3})}^{2}}$

Multiply $2$ by $3$ .

$\frac{16 x^{4} y^{10}}{8 x^{3} y^{6} \cdot 3^{2} {(x^{3})}^{2}}$

Raise $3$ to the power of $2$ .

$\frac{16 x^{4} y^{10}}{8 x^{3} y^{6} \cdot 9 {(x^{3})}^{2}}$

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

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

$\frac{16 x^{4} y^{10}}{8 x^{3} y^{6} \cdot 9 x^{3 \cdot 2}}$

Multiply $3$ by $2$ .

$\frac{16 x^{4} y^{10}}{8 x^{3} y^{6} \cdot 9 x^{6}}$

Combine exponents.

Multiply $9$ by $8$ .

$\frac{16 x^{4} y^{10}}{72 x^{3} y^{6} x^{6}}$

Multiply $x^{3}$ by $x^{6}$ by adding the exponents.

Move $x^{6}$ .

$\frac{16 x^{4} y^{10}}{72 (x^{6} x^{3}) y^{6}}$

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

$\frac{16 x^{4} y^{10}}{72 x^{6 + 3} y^{6}}$

Add $6$ and $3$ .

$\frac{16 x^{4} y^{10}}{72 x^{9} y^{6}}$

Step 3

Reduce the expression by cancelling the common factors.

Cancel the common factor of $16$ and $72$ .

Factor $8$ out of $16 x^{4} y^{10}$ .

$\frac{8 (2 x^{4} y^{10})}{72 x^{9} y^{6}}$

Cancel the common factors.

Factor $8$ out of $72 x^{9} y^{6}$ .

$\frac{8 (2 x^{4} y^{10})}{8 (9 x^{9} y^{6})}$

Cancel the common factor.

$\frac{8 (2 x^{4} y^{10})}{8 (9 x^{9} y^{6})}$

Rewrite the expression.

$\frac{2 x^{4} y^{10}}{9 x^{9} y^{6}}$

Cancel the common factor of $x^{4}$ and $x^{9}$ .

Factor $x^{4}$ out of $2 x^{4} y^{10}$ .

$\frac{x^{4} (2 y^{10})}{9 x^{9} y^{6}}$

Cancel the common factors.

Factor $x^{4}$ out of $9 x^{9} y^{6}$ .

$\frac{x^{4} (2 y^{10})}{x^{4} (9 x^{5} y^{6})}$

Cancel the common factor.

$\frac{x^{4} (2 y^{10})}{x^{4} (9 x^{5} y^{6})}$

Rewrite the expression.

$\frac{2 y^{10}}{9 x^{5} y^{6}}$

Cancel the common factor of $y^{10}$ and $y^{6}$ .

Factor $y^{6}$ out of $2 y^{10}$ .

$\frac{y^{6} (2 y^{4})}{9 x^{5} y^{6}}$

Cancel the common factors.

Factor $y^{6}$ out of $9 x^{5} y^{6}$ .

$\frac{y^{6} (2 y^{4})}{y^{6} (9 x^{5})}$

Cancel the common factor.

$\frac{y^{6} (2 y^{4})}{y^{6} (9 x^{5})}$

Rewrite the expression.

$\frac{2 y^{4}}{9 x^{5}}$

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

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