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

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

Step 1

Move $t^{- 7}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 2

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

Move $t^{7}$ .

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

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

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

Add $7$ and $4$ .

${(\frac{4 s^{5}}{- 2 s^{- 2} t^{11}})}^{3}$

Step 3

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

${(\frac{4 s^{5} s^{2}}{- 2 t^{11}})}^{3}$

Step 4

Multiply $s^{5}$ by $s^{2}$ by adding the exponents.

Move $s^{2}$ .

${(\frac{4 (s^{2} s^{5})}{- 2 t^{11}})}^{3}$

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

${(\frac{4 s^{2 + 5}}{- 2 t^{11}})}^{3}$

Add $2$ and $5$ .

${(\frac{4 s^{7}}{- 2 t^{11}})}^{3}$

Step 5

Cancel the common factor of $4$ and $- 2$ .

Factor $2$ out of $4 s^{7}$ .

${(\frac{2 (2 s^{7})}{- 2 t^{11}})}^{3}$

Cancel the common factors.

Factor $2$ out of $- 2 t^{11}$ .

${(\frac{2 (2 s^{7})}{2 (- t^{11})})}^{3}$

Cancel the common factor.

${(\frac{2 (2 s^{7})}{2 (- t^{11})})}^{3}$

Rewrite the expression.

${(\frac{2 s^{7}}{- t^{11}})}^{3}$

Step 6

Move the negative in front of the fraction.

${(- \frac{2 s^{7}}{t^{11}})}^{3}$

Step 7

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

Apply the product rule to $- \frac{2 s^{7}}{t^{11}}$ .

${(- 1)}^{3} {(\frac{2 s^{7}}{t^{11}})}^{3}$

Apply the product rule to $\frac{2 s^{7}}{t^{11}}$ .

${(- 1)}^{3} \frac{{(2 s^{7})}^{3}}{{(t^{11})}^{3}}$

Apply the product rule to $2 s^{7}$ .

${(- 1)}^{3} \frac{2^{3} {(s^{7})}^{3}}{{(t^{11})}^{3}}$

Step 8

Raise $- 1$ to the power of $3$ .

$- \frac{2^{3} {(s^{7})}^{3}}{{(t^{11})}^{3}}$

Step 9

Simplify the numerator.

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

$- \frac{8 {(s^{7})}^{3}}{{(t^{11})}^{3}}$

Multiply the exponents in ${(s^{7})}^{3}$ .

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

$- \frac{8 s^{7 \cdot 3}}{{(t^{11})}^{3}}$

Multiply $7$ by $3$ .

$- \frac{8 s^{21}}{{(t^{11})}^{3}}$

Step 10

Multiply the exponents in ${(t^{11})}^{3}$ .

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

$- \frac{8 s^{21}}{t^{11 \cdot 3}}$

Multiply $11$ by $3$ .

$- \frac{8 s^{21}}{t^{33}}$

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

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