Solve the Nested Exponent Expression: (16^6)^7

To solve the expression $(16^6)^7$, we will use the power of a power rule for exponents. This rule states that when you raise a power to another power, you multiply the exponents. Here are the steps:

• Identify the components: The base is 16, and the inner exponent is 6. The outer exponent is 7.
• Apply the power of a power rule: According to the rule, $(a^m)^n = a^{m \cdot n}$. Thus, $(16^6)^7 = 16^{6 \cdot 7}$.
• Multiply the exponents: Calculate the product of the exponents $6 \times 7$. This gives us 42.
• Rewrite the expression: Substitute the product back into the expression, giving us $16^{42}$.

Therefore, the simplified expression is $\mathbf{16^{42}}$.

Checking against the answer choices, we find:
1. $16^{42}$ is given as choice 1.
2. Other choices do not match the simplified expression.
Choice 1 is correct because it accurately reflects the application of exponent rules.

Consequently, we conclude that the correct solution is $\mathbf{16^{42}}$.