Simplify (x^2y)^2(4x^3y^7)^3

${(x^{2} y)}^{2} {(4 x^{3} y^{7})}^{3}$

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

${(x^{2})}^{2} y^{2} {(4 x^{3} y^{7})}^{3}$

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

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

$x^{2 \cdot 2} y^{2} {(4 x^{3} y^{7})}^{3}$

Multiply $2$ by $2$ .

$x^{4} y^{2} {(4 x^{3} y^{7})}^{3}$

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

Apply the product rule to $4 x^{3} y^{7}$ .

$x^{4} y^{2} ({(4 x^{3})}^{3} {(y^{7})}^{3})$

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

$x^{4} y^{2} (4^{3} {(x^{3})}^{3} {(y^{7})}^{3})$

Rewrite using the commutative property of multiplication.

$4^{3} x^{4} y^{2} ({(x^{3})}^{3} {(y^{7})}^{3})$

Raise $4$ to the power of $3$ .

$64 x^{4} y^{2} ({(x^{3})}^{3} {(y^{7})}^{3})$

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

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

$64 x^{4} y^{2} (x^{3 \cdot 3} {(y^{7})}^{3})$

Multiply $3$ by $3$ .

$64 x^{4} y^{2} (x^{9} {(y^{7})}^{3})$

Multiply $x^{4}$ by $x^{9}$ by adding the exponents.

Move $x^{9}$ .

$64 (x^{9} x^{4}) y^{2} {(y^{7})}^{3}$

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

$64 x^{9 + 4} y^{2} {(y^{7})}^{3}$

Add $9$ and $4$ .

$64 x^{13} y^{2} {(y^{7})}^{3}$

Multiply the exponents in ${(y^{7})}^{3}$ .

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

$64 x^{13} y^{2} y^{7 \cdot 3}$

Multiply $7$ by $3$ .

$64 x^{13} y^{2} y^{21}$

Multiply $y^{2}$ by $y^{21}$ by adding the exponents.

Move $y^{21}$ .

$64 x^{13} (y^{21} y^{2})$

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

$64 x^{13} y^{21 + 2}$

Add $21$ and $2$ .

$64 x^{13} y^{23}$