Write the Fraction in Simplest Form ((3d^3z)^-3)/(5d^-3z^4)

$\frac{{(3 d^{3} z)}^{- 3}}{5 d^{- 3} z^{4}}$

Step 1

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

$\frac{1}{5 d^{- 3} z^{4} {(3 d^{3} z)}^{3}}$

Step 2

Move $d^{- 3}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

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

Step 3

Simplify the denominator.

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

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

Multiply $z^{4}$ by $z^{3}$ by adding the exponents.

Move $z^{3}$ .

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

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

$\frac{d^{3}}{5 z^{3 + 4} {(3 d^{3})}^{3}}$

Add $3$ and $4$ .

$\frac{d^{3}}{5 z^{7} {(3 d^{3})}^{3}}$

Apply the product rule to $3 d^{3}$ .

$\frac{d^{3}}{5 z^{7} (3^{3} {(d^{3})}^{3})}$

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

$\frac{d^{3}}{5 z^{7} (27 {(d^{3})}^{3})}$

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

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

$\frac{d^{3}}{5 z^{7} (27 d^{3 \cdot 3})}$

Multiply $3$ by $3$ .

$\frac{d^{3}}{5 z^{7} (27 d^{9})}$

$\frac{d^{3}}{5 z^{7} \cdot 27 d^{9}}$

Multiply $27$ by $5$ .

$\frac{d^{3}}{135 z^{7} d^{9}}$

Step 4

Cancel the common factor of $d^{3}$ and $d^{9}$ .

Multiply by $1$ .

$\frac{d^{3} \cdot 1}{135 z^{7} d^{9}}$

Cancel the common factors.

Factor $d^{3}$ out of $135 z^{7} d^{9}$ .

$\frac{d^{3} \cdot 1}{d^{3} (135 z^{7} d^{6})}$

Cancel the common factor.

$\frac{d^{3} \cdot 1}{d^{3} (135 z^{7} d^{6})}$

Rewrite the expression.

$\frac{1}{135 z^{7} d^{6}}$