To solve the expression \(\left(2 x^2+\frac{4}{x}\right)^{12}\) in the form \(a(2)^b\), follow these steps:
1. Simplify the expression inside the parentheses:
[tex]\[
2 x^2 + \frac{4}{x}
\][/tex]
Notice that we can factor out a 2:
[tex]\[
2 x^2 + \frac{4}{x} = 2 \left( x^2 + \frac{2}{x} \right)
\][/tex]
2. Substitute this back into the original expression:
[tex]\[
\left(2 x^2 + \frac{4}{x}\right)^{12} = \left[ 2 \left( x^2 + \frac{2}{x} \right) \right]^{12}
\][/tex]
3. Apply the exponent to both factors inside the parentheses:
Using the property of exponents \((ab)^n = a^n b^n\):
[tex]\[
\left[ 2 \left( x^2 + \frac{2}{x} \right) \right]^{12} = 2^{12} \left( x^2 + \frac{2}{x} \right)^{12}
\][/tex]
4. Identify the constants and terms:
Here, \(2^{12}\) is a constant term, and \(\left( x^2 + \frac{2}{x} \right)^{12}\) is the remaining term.
5. Express the constant \(\mathbf{2^{12}}\) in its form:
The term \(2^{12}\) evaluates to:
[tex]\[
2^{12} = 4096
\][/tex]
6. Combine all results into the desired form \(a (2)^b\):
Given that \(a = 1\), and \(b = 12\), the expression simplifies to:
[tex]\[
a (2)^b = 4096 \cdot \left( x^2 + \frac{2}{x} \right)^{12}
\][/tex]
Thus, the final answer in the specified form is:
[tex]\[
a = 1, \quad b = 12, \quad\text{and the constant term } 2^{12} = 4096.
\][/tex]
Therefore, \(\left(2 x^2 + \frac{4}{x}\right)^{12}\) can be expressed as:
[tex]\[
1 \cdot (2)^{12} = 4096.
\][/tex]
So, the simplified form of the given expression is [tex]\(4096 \left( x^2 + \frac{2}{x} \right)^{12}\)[/tex].
