Use the rules of exponents to simplify the expression.

Question

left parenthesis negative r Superscript 4 Baseline s right parenthesis Superscript 4 Baseline left parenthesis negative r cubed s cubed right parenthesis squared

Answer 1

To simplify the expression \(( -r^4 s )^4 ( -r^3 s^3 )^2\), we will apply the rules of exponents step by step.

1. Simplify each part separately:

   • For \(( -r^4 s )^4\): \[ ( -r^4 s )^4 = (-1)^4 (r^4)^4 (s)^4 = 1 \cdot r^{16} \cdot s^4 = r^{16} s^4 \]

   • For \(( -r^3 s^3 )^2\): \[ ( -r^3 s^3 )^2 = (-1)^2 (r^3)^2 (s^3)^2 = 1 \cdot r^6 \cdot s^6 = r^6 s^6 \]

2. Combine the results: Now we combine the simplified parts: \[ ( -r^4 s )^4 ( -r^3 s^3 )^2 = (r^{16} s^4)(r^6 s^6) \]

3. Use the properties of exponents to combine:

   • For the \(r\) terms: \[ r^{16} \cdot r^6 = r^{16 + 6} = r^{22} \]

   • For the \(s\) terms: \[ s^4 \cdot s^6 = s^{4 + 6} = s^{10} \]

4. Final result: Putting it all together, we have: \[ r^{22} s^{10} \]

Thus, the simplified expression is: \[ \boxed{r^{22} s^{10}} \]