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

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

Step 1

Move ${(x^{2} y^{3})}^{- 2}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 2

Simplify the denominator.

Apply the product rule to $x^{6} y^{3} z$ .

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

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

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

Apply the product rule to $x^{6} y^{3}$ .

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

Multiply the exponents in ${(x^{6})}^{2}$ .

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

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

Multiply $6$ by $2$ .

${(\frac{1}{x^{12} {(y^{3})}^{2} z^{2} {(x^{2})}^{2} {(y^{3})}^{2}})}^{3}$

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

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

${(\frac{1}{x^{12} y^{3 \cdot 2} z^{2} {(x^{2})}^{2} {(y^{3})}^{2}})}^{3}$

Multiply $3$ by $2$ .

${(\frac{1}{x^{12} y^{6} z^{2} {(x^{2})}^{2} {(y^{3})}^{2}})}^{3}$

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

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

${(\frac{1}{x^{12} y^{6} z^{2} x^{2 \cdot 2} {(y^{3})}^{2}})}^{3}$

Multiply $2$ by $2$ .

${(\frac{1}{x^{12} y^{6} z^{2} x^{4} {(y^{3})}^{2}})}^{3}$

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

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

${(\frac{1}{x^{12} y^{6} z^{2} x^{4} y^{3 \cdot 2}})}^{3}$

Multiply $3$ by $2$ .

${(\frac{1}{x^{12} y^{6} z^{2} x^{4} y^{6}})}^{3}$

Combine exponents.

Multiply $x^{12}$ by $x^{4}$ by adding the exponents.

Move $x^{4}$ .

${(\frac{1}{x^{4} x^{12} y^{6} z^{2} y^{6}})}^{3}$

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

${(\frac{1}{x^{4 + 12} y^{6} z^{2} y^{6}})}^{3}$

Add $4$ and $12$ .

${(\frac{1}{x^{16} y^{6} z^{2} y^{6}})}^{3}$

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

Move $y^{6}$ .

${(\frac{1}{x^{16} (y^{6} y^{6}) z^{2}})}^{3}$

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

${(\frac{1}{x^{16} y^{6 + 6} z^{2}})}^{3}$

Add $6$ and $6$ .

${(\frac{1}{x^{16} y^{12} z^{2}})}^{3}$

Step 3

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

Apply the product rule to $\frac{1}{x^{16} y^{12} z^{2}}$ .

$\frac{1^{3}}{{(x^{16} y^{12} z^{2})}^{3}}$

Apply the product rule to $x^{16} y^{12} z^{2}$ .

$\frac{1^{3}}{{(x^{16} y^{12})}^{3} {(z^{2})}^{3}}$

Apply the product rule to $x^{16} y^{12}$ .

$\frac{1^{3}}{{(x^{16})}^{3} {(y^{12})}^{3} {(z^{2})}^{3}}$

Step 4

One to any power is one.

$\frac{1}{{(x^{16})}^{3} {(y^{12})}^{3} {(z^{2})}^{3}}$

Step 5

Simplify the denominator.

Multiply the exponents in ${(x^{16})}^{3}$ .

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

$\frac{1}{x^{16 \cdot 3} {(y^{12})}^{3} {(z^{2})}^{3}}$

Multiply $16$ by $3$ .

$\frac{1}{x^{48} {(y^{12})}^{3} {(z^{2})}^{3}}$

Multiply the exponents in ${(y^{12})}^{3}$ .

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

$\frac{1}{x^{48} y^{12 \cdot 3} {(z^{2})}^{3}}$

Multiply $12$ by $3$ .

$\frac{1}{x^{48} y^{36} {(z^{2})}^{3}}$

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

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

$\frac{1}{x^{48} y^{36} z^{2 \cdot 3}}$

Multiply $2$ by $3$ .

$\frac{1}{x^{48} y^{36} z^{6}}$

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

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