Simplify ((3s^-2t^7)/(6s^3t^-5))^-4

${(\frac{3 s^{- 2} t^{7}}{6 s^{3} t^{- 5}})}^{- 4}$

Step 1

Move $s^{- 2}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

${(\frac{3 t^{7}}{6 s^{3} t^{- 5} s^{2}})}^{- 4}$

Step 2

Multiply $s^{3}$ by $s^{2}$ by adding the exponents.

Move $s^{2}$ .

${(\frac{3 t^{7}}{6 (s^{2} s^{3}) t^{- 5}})}^{- 4}$

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

${(\frac{3 t^{7}}{6 s^{2 + 3} t^{- 5}})}^{- 4}$

Add $2$ and $3$ .

${(\frac{3 t^{7}}{6 s^{5} t^{- 5}})}^{- 4}$

Step 3

Move $t^{- 5}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

${(\frac{3 t^{7} t^{5}}{6 s^{5}})}^{- 4}$

Step 4

Multiply $t^{7}$ by $t^{5}$ by adding the exponents.

Move $t^{5}$ .

${(\frac{3 (t^{5} t^{7})}{6 s^{5}})}^{- 4}$

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

${(\frac{3 t^{5 + 7}}{6 s^{5}})}^{- 4}$

Add $5$ and $7$ .

${(\frac{3 t^{12}}{6 s^{5}})}^{- 4}$

Step 5

Cancel the common factor of $3$ and $6$ .

Factor $3$ out of $3 t^{12}$ .

${(\frac{3 (t^{12})}{6 s^{5}})}^{- 4}$

Cancel the common factors.

Factor $3$ out of $6 s^{5}$ .

${(\frac{3 (t^{12})}{3 (2 s^{5})})}^{- 4}$

Cancel the common factor.

${(\frac{3 t^{12}}{3 (2 s^{5})})}^{- 4}$

Rewrite the expression.

${(\frac{t^{12}}{2 s^{5}})}^{- 4}$

Step 6

Change the sign of the exponent by rewriting the base as its reciprocal.

${(\frac{2 s^{5}}{t^{12}})}^{4}$

Step 7

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

Apply the product rule to $\frac{2 s^{5}}{t^{12}}$ .

$\frac{{(2 s^{5})}^{4}}{{(t^{12})}^{4}}$

Apply the product rule to $2 s^{5}$ .

$\frac{2^{4} {(s^{5})}^{4}}{{(t^{12})}^{4}}$

Step 8

Simplify the numerator.

Raise $2$ to the power of $4$ .

$\frac{16 {(s^{5})}^{4}}{{(t^{12})}^{4}}$

Multiply the exponents in ${(s^{5})}^{4}$ .

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

$\frac{16 s^{5 \cdot 4}}{{(t^{12})}^{4}}$

Multiply $5$ by $4$ .

$\frac{16 s^{20}}{{(t^{12})}^{4}}$

Step 9

Multiply the exponents in ${(t^{12})}^{4}$ .

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

$\frac{16 s^{20}}{t^{12 \cdot 4}}$

Multiply $12$ by $4$ .

$\frac{16 s^{20}}{t^{48}}$

Simplify ((3s^-2t^7)/(6s^3t^-5))^-4

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