Simplify (3xy^-2)^2(2x^3yz)(6yz^2)^-1

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

Step 1

Rewrite using the commutative property of multiplication.

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

Step 2

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 3

Multiply $3 x \frac{1}{y^{2}}$ .

Combine $3$ and $\frac{1}{y^{2}}$ .

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

Combine $x$ and $\frac{3}{y^{2}}$ .

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

Step 4

Move $3$ to the left of $x$ .

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

Step 5

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

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

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

Apply the product rule to $3 x$ .

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

Step 6

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

$2 \frac{9 x^{2}}{{(y^{2})}^{2}} x^{3} y z {(6 y z^{2})}^{- 1}$

Step 7

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

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

$2 \frac{9 x^{2}}{y^{2 \cdot 2}} x^{3} y z {(6 y z^{2})}^{- 1}$

Multiply $2$ by $2$ .

$2 \frac{9 x^{2}}{y^{4}} x^{3} y z {(6 y z^{2})}^{- 1}$

Step 8

Multiply $2 \frac{9 x^{2}}{y^{4}}$ .

Combine $2$ and $\frac{9 x^{2}}{y^{4}}$ .

$\frac{2 (9 x^{2})}{y^{4}} x^{3} y z {(6 y z^{2})}^{- 1}$

Multiply $9$ by $2$ .

$\frac{18 x^{2}}{y^{4}} x^{3} y z {(6 y z^{2})}^{- 1}$

Step 9

Multiply $\frac{18 x^{2}}{y^{4}} x^{3}$ .

Combine $\frac{18 x^{2}}{y^{4}}$ and $x^{3}$ .

$\frac{18 x^{2} x^{3}}{y^{4}} y z {(6 y z^{2})}^{- 1}$

Multiply $x^{2}$ by $x^{3}$ by adding the exponents.

Move $x^{3}$ .

$\frac{18 (x^{3} x^{2})}{y^{4}} y z {(6 y z^{2})}^{- 1}$

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

$\frac{18 x^{3 + 2}}{y^{4}} y z {(6 y z^{2})}^{- 1}$

Add $3$ and $2$ .

$\frac{18 x^{5}}{y^{4}} y z {(6 y z^{2})}^{- 1}$

Step 10

Cancel the common factor of $y$ .

Factor $y$ out of $y^{4}$ .

$\frac{18 x^{5}}{y \cdot y^{3}} y z {(6 y z^{2})}^{- 1}$

Cancel the common factor.

$\frac{18 x^{5}}{y \cdot y^{3}} y z {(6 y z^{2})}^{- 1}$

Rewrite the expression.

$\frac{18 x^{5}}{y^{3}} z {(6 y z^{2})}^{- 1}$

Step 11

Combine $\frac{18 x^{5}}{y^{3}}$ and $z$ .

$\frac{18 x^{5} z}{y^{3}} {(6 y z^{2})}^{- 1}$

Step 12

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{18 x^{5} z}{y^{3}} \cdot \frac{1}{6 y z^{2}}$

Step 13

Combine.

$\frac{18 x^{5} z \cdot 1}{y^{3} (6 y z^{2})}$

Step 14

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

Move $y$ .

$\frac{18 x^{5} z \cdot 1}{y \cdot y^{3} (6 z^{2})}$

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

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

$\frac{18 x^{5} z \cdot 1}{y^{1} y^{3} (6 z^{2})}$

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

$\frac{18 x^{5} z \cdot 1}{y^{1 + 3} (6 z^{2})}$

Add $1$ and $3$ .

$\frac{18 x^{5} z \cdot 1}{y^{4} (6 z^{2})}$

Step 15

Multiply $18 x^{5} z$ by $1$ .

$\frac{18 x^{5} z}{y^{4} (6 z^{2})}$

Step 16

Cancel the common factor of $18$ and $6$ .

Factor $6$ out of $18 x^{5} z$ .

$\frac{6 (3 x^{5} z)}{y^{4} (6 z^{2})}$

Cancel the common factors.

Factor $6$ out of $y^{4} (6 z^{2})$ .

$\frac{6 (3 x^{5} z)}{6 (y^{4} (z^{2}))}$

Cancel the common factor.

$\frac{6 (3 x^{5} z)}{6 (y^{4} (z^{2}))}$

Rewrite the expression.

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

Step 17

Cancel the common factor of $z$ and $z^{2}$ .

Factor $z$ out of $3 x^{5} z$ .

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

Cancel the common factors.

Factor $z$ out of $y^{4} (z^{2})$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

Simplify (3xy^-2)^2(2x^3yz)(6yz^2)^-1

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