Simplify (3x^5y^-2z)^3

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

Step 1

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

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

Step 2

Multiply $3 x^{5} \frac{1}{y^{2}}$ .

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

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

Combine $x^{5}$ and $\frac{3}{y^{2}}$ .

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

Step 3

Move $3$ to the left of $x^{5}$ .

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

Step 4

Combine $\frac{3 x^{5}}{y^{2}}$ and $z$ .

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

Step 5

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

Apply the product rule to $\frac{3 x^{5} z}{y^{2}}$ .

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

Apply the product rule to $3 x^{5} z$ .

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

Apply the product rule to $3 x^{5}$ .

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

Step 6

Simplify the numerator.

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

$\frac{27 {(x^{5})}^{3} z^{3}}{{(y^{2})}^{3}}$

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

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

$\frac{27 x^{5 \cdot 3} z^{3}}{{(y^{2})}^{3}}$

Multiply $5$ by $3$ .

$\frac{27 x^{15} z^{3}}{{(y^{2})}^{3}}$

Step 7

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

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

$\frac{27 x^{15} z^{3}}{y^{2 \cdot 3}}$

Multiply $2$ by $3$ .

$\frac{27 x^{15} z^{3}}{y^{6}}$