Simplify ((by)^8)^9: Evaluating Compound Exponent Expressions

Insert the corresponding expression:

$\left(\left(by\right)^8\right)^9=$

To solve this problem, we'll apply the power of a power rule for exponents. The rule states that if you have an expression of the form $(x^m)^n$, it simplifies to $x^{m \cdot n}$.

• Step 1: Identify the given expression: $\left(\left(by\right)^8\right)^9$.
• Step 2: Apply the power of a power rule by multiplying the exponents.

Let's work through the solution:
Step 1: We start with $\left(\left(by\right)^8\right)^9$. Here, $\left(by\right)$ is considered as a single base.
Step 2: Apply the power of a power rule: $(by)^{8 \cdot 9}$.
Step 3: Calculate the exponent multiplication: $8 \times 9 = 72$.

Therefore, the simplified expression is $(b \times y)^{72}$.

Analyzing the choices provided:

• Choice 1: $(b \times y)^{17}$ - Incorrect because the exponents should multiply to 72.
• Choice 2: $(b \times y)^1$ - Incorrect because it does not reflect the multiplication of exponents.
• Choice 3: $(b \times y)^{\frac{9}{8}}$ - Incorrect, involves incorrect operations on the exponents.
• Choice 4: $(b \times y)^{72}$ - Correct as it correctly applies the power of a power rule.

Thus, the correct answer is Choice 4: $\left(b \times y\right)^{72}$.