Question

a45 - exponents power rule
score: 4/10 answered: 4/10
question 5
simplify the expression ((3a^{3}b^{4}c^{4})^{2}(2a^{3}b^{4}c^{4})^{3})
question help: message instructor

Explanation:

Step1: Apply Power of a Product Rule

For \((3a^{3}b^{4}c^{4})^{2}\), use \((xy)^n = x^n y^n\):
\(3^{2}(a^{3})^{2}(b^{4})^{2}(c^{4})^{2}=9a^{6}b^{8}c^{8}\)

For \((2a^{3}b^{4}c^{4})^{3}\), use \((xy)^n = x^n y^n\):
\(2^{3}(a^{3})^{3}(b^{4})^{3}(c^{4})^{3}=8a^{9}b^{12}c^{12}\)

Step2: Multiply the Two Results

Multiply \(9a^{6}b^{8}c^{8}\) and \(8a^{9}b^{12}c^{12}\):

• Coefficients: \(9 \times 8 = 72\)
• For \(a\): \(a^{6} \times a^{9}=a^{6 + 9}=a^{15}\) (using \(x^m \times x^n = x^{m + n}\))
• For \(b\): \(b^{8} \times b^{12}=b^{8 + 12}=b^{20}\)
• For \(c\): \(c^{8} \times c^{12}=c^{8 + 12}=c^{20}\)

Combine them: \(72a^{15}b^{20}c^{20}\)

Answer:

\(72a^{15}b^{20}c^{20}\)