Question
Simplify: $\left(3 x^{3}\right)^{3}\left(x^{5}\right)^{4}$.

Provide your answer below:

Step 1 ：The given expression is a product of two terms, both raised to a power. To simplify this, we can use the rule of exponents that states \((a^m)^n = a^{mn}\), where \(a\) is the base and \(m\) and \(n\) are the exponents. This rule allows us to multiply the exponents when a power is raised to another power.

Step 2 ：Applying this rule to the first term \((3 x^{3})^{3}\), we get \(3^{3} * x^{3*3} = 27x^{9}\).

Step 3 ：Applying this rule to the second term \((x^{5})^{4}\), we get \(x^{5*4} = x^{20}\).

Step 4 ：We can then use another rule of exponents that states \(a^m * a^n = a^{m+n}\), where \(a\) is the base and \(m\) and \(n\) are the exponents. This rule allows us to add the exponents when the same base is multiplied.

Step 5 ：Applying this rule to the terms \(27x^{9}\) and \(x^{20}\), we get \(27x^{9+20} = 27x^{29}\).

Step 6 ：So, the simplified form of the given expression is \(\boxed{27x^{29}}\).