Simplify the Nested Expression: ((by)^7)^6

To solve this problem, we'll implement the following steps:

• Step 1: Identify the initial expression that needs simplification: $\left(\left(by\right)^7\right)^6$.
• Step 2: Apply the "power of a power" rule, which states $(x^m)^n = x^{m \times n}$.
• Step 3: Calculate the product of the exponents $7$ and $6$.

Let's explore each step in detail:
Step 1: We begin with the expression $\left(\left(by\right)^7\right)^6$. This indicates $(by)^7$ is raised to another power, 6.

Step 2: According to the power of a power rule, we multiply the exponents:

Step 3: The resulting expression is simplified and expressed as $(b \times y)^{42}$.

Therefore, the solution to the problem is $\left(b \times y\right)^{42}$.

Now, comparing our result with the provided choice options:

• Choice 1: $\left(b\times y\right)^{7+6}$ - Incorrect, because it adds the exponents, which violates the power of a power rule.
• Choice 2: $\left(b\times y\right)^{7\times6}$ - Correct, as it reflects the calculated exponent multiplication: $42$.
• Choice 3: $\left(b\times y\right)^{7-6}$ - Incorrect, because it subtracts the exponents, irrelevant in this context.
• Choice 4: $\left(b\times y\right)^{\frac{6}{7}}$ - Incorrect, as it divides the exponents, not applicable here.

Thus, the correct answer is indeed Choice 2.