Simplify Nested Exponents: ((6×9)^x)^a Expression Challenge

To solve this problem, we'll follow these steps:

• Step 1: Identify the given expression and its structure
• Step 2: Apply the power of a power rule of exponents
• Step 3: Simplify the expression and compare it with the given choices

Now, let's work through each step:

Step 1: We are given the expression $\left(\left(6\times9\right)^x\right)^a$. The base of this expression is $6 \times 9$, while the exponents are nested with $x$ in the first power and $a$ in the outer power.

Step 2: According to the power of a power rule, we have:
$\left(b^m\right)^n = b^{m \times n}$
Here, replace $b$ with $(6 \times 9)$, $m$ with $x$, and $n$ with $a$. Therefore:
$\left(\left(6\times9\right)^x\right)^a = \left(6\times9\right)^{x \times a}$

Step 3: Simplify and compare against the available choices:

• Choice 1: $\left(6\times9\right)^{xa}$
• Choice 2: $\left(6\times9\right)^{x+a}$
• Choice 3: $\left(6\times9\right)^{x-a}$
• Choice 4: $\left(6\times9\right)^{\frac{a}{x}}$

The simplified expression $\left(6\times9\right)^{xa}$ matches Choice 1.

Therefore, the correct answer is Choice 1: $\left(6\times9\right)^{xa}$.