Simplify -(-xy^2z^3)^5(x^4yz)^2

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

Step 1

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

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

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

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

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

Apply the product rule to $- x$ .

$- ({(- 1)}^{5} x^{5} {(y^{2})}^{5} {(z^{3})}^{5}) {(x^{4} y z)}^{2}$

Step 2

Multiply $- 1$ by ${(- 1)}^{5}$ by adding the exponents.

Move ${(- 1)}^{5}$ .

${(- 1)}^{5} \cdot - 1 (x^{5} {(y^{2})}^{5} {(z^{3})}^{5}) {(x^{4} y z)}^{2}$

Multiply ${(- 1)}^{5}$ by $- 1$ .

Raise $- 1$ to the power of $1$ .

${(- 1)}^{5} \cdot {(- 1)}^{1} (x^{5} {(y^{2})}^{5} {(z^{3})}^{5}) {(x^{4} y z)}^{2}$

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

${(- 1)}^{5 + 1} (x^{5} {(y^{2})}^{5} {(z^{3})}^{5}) {(x^{4} y z)}^{2}$

Add $5$ and $1$ .

${(- 1)}^{6} (x^{5} {(y^{2})}^{5} {(z^{3})}^{5}) {(x^{4} y z)}^{2}$

Step 3

Raise $- 1$ to the power of $6$ .

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

Step 4

Multiply $x^{5} {(y^{2})}^{5} {(z^{3})}^{5}$ by $1$ .

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

Step 5

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

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

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

Multiply $2$ by $5$ .

$x^{5} y^{10} {(z^{3})}^{5} {(x^{4} y z)}^{2}$

Step 6

Multiply the exponents in ${(z^{3})}^{5}$ .

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

$x^{5} y^{10} z^{3 \cdot 5} {(x^{4} y z)}^{2}$

Multiply $3$ by $5$ .

$x^{5} y^{10} z^{15} {(x^{4} y z)}^{2}$

Step 7

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

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

$x^{5} y^{10} z^{15} ({(x^{4} y)}^{2} z^{2})$

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

$x^{5} y^{10} z^{15} ({(x^{4})}^{2} y^{2} z^{2})$

Step 8

Multiply $y^{10}$ by $y^{2}$ by adding the exponents.

Move $y^{2}$ .

$x^{5} (y^{2} y^{10}) z^{15} ({(x^{4})}^{2} z^{2})$

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

$x^{5} y^{2 + 10} z^{15} ({(x^{4})}^{2} z^{2})$

Add $2$ and $10$ .

$x^{5} y^{12} z^{15} ({(x^{4})}^{2} z^{2})$

Step 9

Multiply $z^{15}$ by $z^{2}$ by adding the exponents.

Move $z^{2}$ .

$x^{5} y^{12} (z^{2} z^{15}) {(x^{4})}^{2}$

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

$x^{5} y^{12} z^{2 + 15} {(x^{4})}^{2}$

Add $2$ and $15$ .

$x^{5} y^{12} z^{17} {(x^{4})}^{2}$

Step 10

Multiply the exponents in ${(x^{4})}^{2}$ .

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

$x^{5} y^{12} z^{17} x^{4 \cdot 2}$

Multiply $4$ by $2$ .

$x^{5} y^{12} z^{17} x^{8}$

Step 11

Multiply $x^{5}$ by $x^{8}$ by adding the exponents.

Move $x^{8}$ .

$x^{8} x^{5} y^{12} z^{17}$

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

$x^{8 + 5} y^{12} z^{17}$

Add $8$ and $5$ .

$x^{13} y^{12} z^{17}$

Simplify -(-xy^2z^3)^5(x^4yz)^2

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