Simplify ((-2x^3y^8z^-2)/(6x^-1y^3z^-1))^0

${(\frac{- 2 x^{3} y^{8} z^{- 2}}{6 x^{- 1} y^{3} z^{- 1}})}^{0}$

Step 1

Move $z^{- 2}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

${(\frac{- 2 x^{3} y^{8}}{6 x^{- 1} y^{3} z^{- 1} z^{2}})}^{0}$

Step 2

Simplify the denominator.

Multiply $z^{- 1}$ by $z^{2}$ by adding the exponents.

Move $z^{2}$ .

${(\frac{- 2 x^{3} y^{8}}{6 x^{- 1} y^{3} (z^{2} z^{- 1})})}^{0}$

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

${(\frac{- 2 x^{3} y^{8}}{6 x^{- 1} y^{3} z^{2 - 1}})}^{0}$

Subtract $1$ from $2$ .

${(\frac{- 2 x^{3} y^{8}}{6 x^{- 1} y^{3} z^{1}})}^{0}$

Simplify $6 x^{- 1} y^{3} z^{1}$ .

${(\frac{- 2 x^{3} y^{8}}{6 x^{- 1} y^{3} z})}^{0}$

Step 3

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

${(\frac{- 2 x^{3} y^{8} x}{6 y^{3} z})}^{0}$

Step 4

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

Move $x$ .

${(\frac{- 2 (x \cdot x^{3}) y^{8}}{6 y^{3} z})}^{0}$

Multiply $x$ by $x^{3}$ .

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

${(\frac{- 2 (x^{1} x^{3}) y^{8}}{6 y^{3} z})}^{0}$

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

${(\frac{- 2 x^{1 + 3} y^{8}}{6 y^{3} z})}^{0}$

Add $1$ and $3$ .

${(\frac{- 2 x^{4} y^{8}}{6 y^{3} z})}^{0}$

Step 5

Cancel the common factor of $- 2$ and $6$ .

Factor $2$ out of $- 2 x^{4} y^{8}$ .

${(\frac{2 (- x^{4} y^{8})}{6 y^{3} z})}^{0}$

Cancel the common factors.

Factor $2$ out of $6 y^{3} z$ .

${(\frac{2 (- x^{4} y^{8})}{2 (3 y^{3} z)})}^{0}$

Cancel the common factor.

${(\frac{2 (- x^{4} y^{8})}{2 (3 y^{3} z)})}^{0}$

Rewrite the expression.

${(\frac{- x^{4} y^{8}}{3 y^{3} z})}^{0}$

Step 6

Cancel the common factor of $y^{8}$ and $y^{3}$ .

Factor $y^{3}$ out of $- x^{4} y^{8}$ .

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 7

Move the negative in front of the fraction.

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

Step 8

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

Apply the product rule to $- \frac{x^{4} y^{5}}{3 z}$ .

${(- 1)}^{0} {(\frac{x^{4} y^{5}}{3 z})}^{0}$

Apply the product rule to $\frac{x^{4} y^{5}}{3 z}$ .

${(- 1)}^{0} \frac{{(x^{4} y^{5})}^{0}}{{(3 z)}^{0}}$

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

${(- 1)}^{0} \frac{{(x^{4})}^{0} {(y^{5})}^{0}}{{(3 z)}^{0}}$

Apply the product rule to $3 z$ .

${(- 1)}^{0} \frac{{(x^{4})}^{0} {(y^{5})}^{0}}{3^{0} z^{0}}$

Step 9

Simplify the expression.

Anything raised to $0$ is $1$ .

$1 \frac{{(x^{4})}^{0} {(y^{5})}^{0}}{3^{0} z^{0}}$

Multiply $\frac{{(x^{4})}^{0} {(y^{5})}^{0}}{3^{0} z^{0}}$ by $1$ .

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

Step 10

Simplify the numerator.

Multiply the exponents in ${(x^{4})}^{0}$ .

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

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

Multiply $4$ by $0$ .

$\frac{x^{0} {(y^{5})}^{0}}{3^{0} z^{0}}$

Anything raised to $0$ is $1$ .

$\frac{1 {(y^{5})}^{0}}{3^{0} z^{0}}$

Multiply the exponents in ${(y^{5})}^{0}$ .

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

$\frac{1 y^{5 \cdot 0}}{3^{0} z^{0}}$

Multiply $5$ by $0$ .

$\frac{1 y^{0}}{3^{0} z^{0}}$

Anything raised to $0$ is $1$ .

$\frac{1 \cdot 1}{3^{0} z^{0}}$

Multiply $1$ by $1$ .

$\frac{1}{3^{0} z^{0}}$

Step 11

Simplify the denominator.

Anything raised to $0$ is $1$ .

$\frac{1}{1 z^{0}}$

Anything raised to $0$ is $1$ .

$\frac{1}{1 \cdot 1}$

Multiply $1$ by $1$ .

$\frac{1}{1}$

Step 12

Divide $1$ by $1$ .

$1$

Simplify ((-2x^3y^8z^-2)/(6x^-1y^3z^-1))^0

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