Question

In Exercises \(5-8,\) rewrite the exponential expression to have the indicated base. \(9^{2 x}, \quad\) base 3

Short answer

The expression \(9^{2x}\) can be rewritten as \(3^{4x}\)

Step-by-step solution

Recognize the Common Base

Observe that 9 is in fact a power of 3 (i.e., \(9 = 3^2\)). This common base will allow us to write the original exponential expression and the new base in terms of this common base. So, we can rewrite \(9^{2x}\) as \((3^2)^{2x}\)

Apply the power of a power rule

The power of a power rule in exponents says that when raising a power to a power, you multiply the exponents. Thus, we can simplify \((3^2)^{2x}\) to become \(3^{2 * 2x}\), which simplifies further to \(3^{4x}\)

Check the Result

As a final step, always check the new expression to ensure it is written in terms of the desired new base. Here, the expression \(3^{4x}\) is indeed written with base 3, which is what was asked for.