Question

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

Short answer

The simplified expression is \(a^{7}b^{5}c^{6}\)

Step-by-step solution

Apply Power to Power Rule

We can begin by applying the power-to-power rule, which states that \( (a^m)^n = a^{m*n} \). This rule needs to be applied to both \( (abc^2)^3 \) and \( (a^2b)^2 \). So we have \( a^{3}*b^{3}*c^{6} \) and \( a^{4}*b^{2} \).

Apply Product of Powers Rule

Next, apply the product of powers rule which states that when multiplying with the same base, add the exponents. Applying the rule to \( a^{3}*a^{4} \) and \( b^{3}*b^{2} \), we get it simplified to \( a^{7} \) and \( b^{5} \). The term \( c^{6} \) remains as it is because there is no other term with the base 'c' to combine with.

Combine the Powers

Combine all three terms to get the final answer. The simplified expression becomes \(a^{7}\)*\(b^{5}\)*\(c^{6}\).