Question

simplify the expression using the properties of exponents.
\\((3ma^2)^3(3m^2a)^2\\)
show your work here
hint: to add an exponent (\\(x^y\\)), type \exponent\ or press \^\

Explanation:

Step1: Apply power of a product rule

For \((3ma^{2})^{3}\), use \((xyz)^n = x^n y^n z^n\). So \((3ma^{2})^{3}=3^{3}m^{3}(a^{2})^{3}\). Calculate \(3^{3}=27\) and \((a^{2})^{3}=a^{2\times3}=a^{6}\), so it becomes \(27m^{3}a^{6}\).

For \((3m^{2}a)^{2}\), use the same rule: \((3m^{2}a)^{2}=3^{2}(m^{2})^{2}a^{2}\). Calculate \(3^{2}=9\) and \((m^{2})^{2}=m^{2\times2}=m^{4}\), so it becomes \(9m^{4}a^{2}\).

Now the expression is \(27m^{3}a^{6}\times9m^{4}a^{2}\).

Step2: Multiply coefficients and use product of powers rule

Multiply coefficients: \(27\times9 = 243\).

For variables with \(m\): use \(x^m\times x^n=x^{m + n}\), so \(m^{3}\times m^{4}=m^{3 + 4}=m^{7}\).

For variables with \(a\): \(a^{6}\times a^{2}=a^{6+2}=a^{8}\).

Combine them: \(243m^{7}a^{8}\).

Answer:

\(243m^{7}a^{8}\)