Simplify ((3m^2t^5)/(2m^3t^-1))^2

${(\frac{3 m^{2} t^{5}}{2 m^{3} t^{- 1}})}^{2}$

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

${(\frac{3 m^{2} t^{5} t}{2 m^{3}})}^{2}$

Multiply $t^{5}$ by $t$ by adding the exponents.

Move $t$ .

${(\frac{3 m^{2} (t \cdot t^{5})}{2 m^{3}})}^{2}$

Multiply $t$ by $t^{5}$ .

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

${(\frac{3 m^{2} (t^{1} t^{5})}{2 m^{3}})}^{2}$

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

${(\frac{3 m^{2} t^{1 + 5}}{2 m^{3}})}^{2}$

Add $1$ and $5$ .

${(\frac{3 m^{2} t^{6}}{2 m^{3}})}^{2}$

Cancel the common factor of $m^{2}$ and $m^{3}$ .

Factor $m^{2}$ out of $3 m^{2} t^{6}$ .

${(\frac{m^{2} (3 t^{6})}{2 m^{3}})}^{2}$

Cancel the common factors.

Factor $m^{2}$ out of $2 m^{3}$ .

${(\frac{m^{2} (3 t^{6})}{m^{2} (2 m)})}^{2}$

Cancel the common factor.

${(\frac{m^{2} (3 t^{6})}{m^{2} (2 m)})}^{2}$

Rewrite the expression.

${(\frac{3 t^{6}}{2 m})}^{2}$

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

Apply the product rule to $\frac{3 t^{6}}{2 m}$ .

$\frac{{(3 t^{6})}^{2}}{{(2 m)}^{2}}$

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

$\frac{3^{2} {(t^{6})}^{2}}{{(2 m)}^{2}}$

Apply the product rule to $2 m$ .

$\frac{3^{2} {(t^{6})}^{2}}{2^{2} m^{2}}$

Simplify the numerator.

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

$\frac{9 {(t^{6})}^{2}}{2^{2} m^{2}}$

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

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

$\frac{9 t^{6 \cdot 2}}{2^{2} m^{2}}$

Multiply $6$ by $2$ .

$\frac{9 t^{12}}{2^{2} m^{2}}$

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

$\frac{9 t^{12}}{4 m^{2}}$