Simplify (x^3y^2z)^3(3xy^2z^3)^2(-3x)^2

${(x^{3} y^{2} z)}^{3} {(3 x y^{2} z^{3})}^{2} {(- 3 x)}^{2}$

Step 1

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

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

${(x^{3} y^{2})}^{3} z^{3} {(3 x y^{2} z^{3})}^{2} {(- 3 x)}^{2}$

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

${(x^{3})}^{3} {(y^{2})}^{3} z^{3} {(3 x y^{2} z^{3})}^{2} {(- 3 x)}^{2}$

Step 2

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

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

$x^{3 \cdot 3} {(y^{2})}^{3} z^{3} {(3 x y^{2} z^{3})}^{2} {(- 3 x)}^{2}$

Multiply $3$ by $3$ .

$x^{9} {(y^{2})}^{3} z^{3} {(3 x y^{2} z^{3})}^{2} {(- 3 x)}^{2}$

Step 3

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

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

$x^{9} y^{2 \cdot 3} z^{3} {(3 x y^{2} z^{3})}^{2} {(- 3 x)}^{2}$

Multiply $2$ by $3$ .

$x^{9} y^{6} z^{3} {(3 x y^{2} z^{3})}^{2} {(- 3 x)}^{2}$

Step 4

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

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

$x^{9} y^{6} z^{3} ({(3 x y^{2})}^{2} {(z^{3})}^{2}) {(- 3 x)}^{2}$

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

$x^{9} y^{6} z^{3} ({(3 x)}^{2} {(y^{2})}^{2} {(z^{3})}^{2}) {(- 3 x)}^{2}$

Apply the product rule to $3 x$ .

$x^{9} y^{6} z^{3} (3^{2} x^{2} {(y^{2})}^{2} {(z^{3})}^{2}) {(- 3 x)}^{2}$

Step 5

Multiply $x^{9}$ by $x^{2}$ by adding the exponents.

Move $x^{2}$ .

$x^{2} x^{9} y^{6} z^{3} (3^{2} {(y^{2})}^{2} {(z^{3})}^{2}) {(- 3 x)}^{2}$

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

$x^{2 + 9} y^{6} z^{3} (3^{2} {(y^{2})}^{2} {(z^{3})}^{2}) {(- 3 x)}^{2}$

Add $2$ and $9$ .

$x^{11} y^{6} z^{3} (3^{2} {(y^{2})}^{2} {(z^{3})}^{2}) {(- 3 x)}^{2}$

Step 6

Rewrite using the commutative property of multiplication.

$3^{2} x^{11} y^{6} z^{3} ({(y^{2})}^{2} {(z^{3})}^{2}) {(- 3 x)}^{2}$

Step 7

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

$9 x^{11} y^{6} z^{3} ({(y^{2})}^{2} {(z^{3})}^{2}) {(- 3 x)}^{2}$

Step 8

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

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

$9 x^{11} y^{6} z^{3} (y^{2 \cdot 2} {(z^{3})}^{2}) {(- 3 x)}^{2}$

Multiply $2$ by $2$ .

$9 x^{11} y^{6} z^{3} (y^{4} {(z^{3})}^{2}) {(- 3 x)}^{2}$

Step 9

Multiply $y^{6}$ by $y^{4}$ by adding the exponents.

Move $y^{4}$ .

$9 x^{11} (y^{4} y^{6}) z^{3} {(z^{3})}^{2} {(- 3 x)}^{2}$

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

$9 x^{11} y^{4 + 6} z^{3} {(z^{3})}^{2} {(- 3 x)}^{2}$

Add $4$ and $6$ .

$9 x^{11} y^{10} z^{3} {(z^{3})}^{2} {(- 3 x)}^{2}$

Step 10

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

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

$9 x^{11} y^{10} z^{3} z^{3 \cdot 2} {(- 3 x)}^{2}$

Multiply $3$ by $2$ .

$9 x^{11} y^{10} z^{3} z^{6} {(- 3 x)}^{2}$

Step 11

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

Move $z^{6}$ .

$9 x^{11} y^{10} (z^{6} z^{3}) {(- 3 x)}^{2}$

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

$9 x^{11} y^{10} z^{6 + 3} {(- 3 x)}^{2}$

Add $6$ and $3$ .

$9 x^{11} y^{10} z^{9} {(- 3 x)}^{2}$

Step 12

Apply the product rule to $- 3 x$ .

$9 x^{11} y^{10} z^{9} ({(- 3)}^{2} x^{2})$

Step 13

Multiply $x^{11}$ by $x^{2}$ by adding the exponents.

Move $x^{2}$ .

$9 (x^{2} x^{11}) y^{10} z^{9} {(- 3)}^{2}$

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

$9 x^{2 + 11} y^{10} z^{9} {(- 3)}^{2}$

Add $2$ and $11$ .

$9 x^{13} y^{10} z^{9} {(- 3)}^{2}$

Step 14

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

$9 x^{13} y^{10} z^{9} \cdot 9$

Step 15

Multiply $9$ by $9$ .

$81 x^{13} y^{10} z^{9}$

Simplify (x^3y^2z)^3(3xy^2z^3)^2(-3x)^2

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