Question

simplify the expression ((2b^{3}c^{4})^{2}). write the variables in alphabetical order.

Explanation:

Step1: Apply the power of a product rule

The power of a product rule states that \((ab)^n = a^n b^n\). So for \((2b^{3}c^{4})^{2}\), we apply this rule to each factor inside the parentheses.
\[
(2b^{3}c^{4})^{2}=2^{2}\times(b^{3})^{2}\times(c^{4})^{2}
\]

Step2: Simplify each term

• Simplify \(2^{2}\): \(2^{2} = 4\)
• Simplify \((b^{3})^{2}\) using the power of a power rule \((a^m)^n=a^{mn}\): \((b^{3})^{2}=b^{3\times2}=b^{6}\)
• Simplify \((c^{4})^{2}\) using the power of a power rule: \((c^{4})^{2}=c^{4\times2}=c^{8}\)

Step3: Combine the simplified terms

Multiply the simplified terms together: \(4\times b^{6}\times c^{8}=4b^{6}c^{8}\) (since we need to write variables in alphabetical order, \(b\) comes before \(c\))

Answer:

\(4b^{6}c^{8}\)