Question

Write your answer as a power or as a product of powers. $$ (-r s)\left(r s^{3}\right)^{2} $$

Short answer

The simplified form of the expression \( (-r s)\left(r s^{3}\right)^{2} \) is \( -r^{3} s^{7} \).

Step-by-step solution

Identify the expression

The expression given is \( (-r s)\left(r s^{3}\right)^{2} \).

Apply power-of-a-product rule

The power-of-a-product rule states that \((ab)^n = a^n * b^n\). Apply this rule to \( (r s^{3})^{2} \) in the equation, resulting in \( (-r s) (r^{2} (s^{3})^{2}) \).

Apply power-to-power rule

The power-to-power rule states \( (a^{n})^{m} = a^{n*m} \). Apply this rule to \( (s^{3})^{2} = s^{^{3*2}} = s^{6} \), leading to \( (-r s) (r^{2} s^{6}) \) in the original expression.

Multiply-term together

Multiply \( -r \) with \( r^{2} \) to get \( -r^{3} \), and \( s \) with \( s^{6} \) to get \( s^{7} \). Therefore, the simplified form of the original expression is \( -r^{3} s^{7} \)