Simplify ((3rt)^2u^-4)/(r^-1t^2u^7)

$\frac{{(3 r t)}^{2} u^{- 4}}{r^{- 1} t^{2} u^{7}}$

Move $u^{- 4}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{{(3 r t)}^{2}}{r^{- 1} t^{2} u^{7} u^{4}}$

Multiply $u^{7}$ by $u^{4}$ by adding the exponents.

Move $u^{4}$ .

$\frac{{(3 r t)}^{2}}{r^{- 1} t^{2} (u^{4} u^{7})}$

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

$\frac{{(3 r t)}^{2}}{r^{- 1} t^{2} u^{4 + 7}}$

Add $4$ and $7$ .

$\frac{{(3 r t)}^{2}}{r^{- 1} t^{2} u^{11}}$

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

$\frac{{(3 r t)}^{2} r}{t^{2} u^{11}}$

Simplify the numerator.

Apply the product rule to $3 r t$ .

$\frac{{(3 r)}^{2} t^{2} r}{t^{2} u^{11}}$

Apply the product rule to $3 r$ .

$\frac{3^{2} r^{2} t^{2} r}{t^{2} u^{11}}$

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

$\frac{9 r^{2} t^{2} r}{t^{2} u^{11}}$

Combine exponents.

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

$\frac{9 (r^{1} r^{2}) t^{2}}{t^{2} u^{11}}$

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

$\frac{9 r^{1 + 2} t^{2}}{t^{2} u^{11}}$

Add $1$ and $2$ .

$\frac{9 r^{3} t^{2}}{t^{2} u^{11}}$

Cancel the common factor of $t^{2}$ .

Cancel the common factor.

$\frac{9 r^{3} t^{2}}{t^{2} u^{11}}$

Rewrite the expression.

$\frac{9 r^{3}}{u^{11}}$