Question

simplify.
$(3a^2b^3)^4$
write your answer without parentheses.

Explanation:

Step1: Apply power of a product rule

The power of a product rule states that \((xy)^n = x^n y^n\). So for \((3a^{2}b^{3})^{4}\), we can apply this rule to each factor inside the parentheses: \(3^{4}\times(a^{2})^{4}\times(b^{3})^{4}\).

Step2: Simplify each term

• For the coefficient: \(3^{4}=3\times3\times3\times3 = 81\).
• For the \(a\) term: Using the power of a power rule \((x^m)^n=x^{mn}\), we have \((a^{2})^{4}=a^{2\times4}=a^{8}\).
• For the \(b\) term: Similarly, \((b^{3})^{4}=b^{3\times4}=b^{12}\).

Step3: Combine the terms

Multiply the simplified terms together: \(81\times a^{8}\times b^{12}=81a^{8}b^{12}\).

Answer:

\(81a^{8}b^{12}\)