Question

simplify
$(3a^{3}b)^{2}(4ab^{2})^{3}$
$?a^{\square}b^{\square}$

Explanation:

Step1: Simplify \((3a^{3}b)^{2}\)

Using the power of a product rule \((xy)^n = x^n y^n\) and power of a power rule \((x^m)^n = x^{mn}\), we have:
\((3a^{3}b)^{2}=3^{2}(a^{3})^{2}b^{2}=9a^{6}b^{2}\)

Step2: Simplify \((4ab^{2})^{3}\)

Using the same rules as above:
\((4ab^{2})^{3}=4^{3}a^{3}(b^{2})^{3}=64a^{3}b^{6}\)

Step3: Multiply the two simplified expressions

Multiply the coefficients and use the product of powers rule \(x^m \cdot x^n = x^{m + n}\) for the variables:
\((9a^{6}b^{2})(64a^{3}b^{6})=(9\times64)a^{6 + 3}b^{2+6}\)
Calculate \(9\times64 = 576\), and \(6 + 3=9\), \(2 + 6 = 8\). So we get \(576a^{9}b^{8}\)

Answer:

\(576a^{9}b^{8}\) (So the coefficient is 576, the exponent of \(a\) is 9, and the exponent of \(b\) is 8)