Solve Nested Exponents: Simplifying (8^5)^10 Using Power Rules

To simplify the expression $\left(8^5\right)^{10}$, we'll apply the power of a power rule for exponents.

• Step 1: Identify the given expression.
• Step 2: Apply the power of a power rule, which states that $(a^m)^n = a^{m \cdot n}$.
• Step 3: Multiply the exponents to simplify the expression.

Now, let's work through each step:
Step 1: The expression given is $\left(8^5\right)^{10}$.
Step 2: We will use the power of a power rule: $(a^m)^n = a^{m \cdot n}$.
Step 3: Multiply the exponents: $5 \cdot 10 = 50$.

Thus, the expression simplifies to $8^{50}$.

The correct simplified form of the expression $\left(8^5\right)^{10}$ is $8^{50}$, which corresponds to choice 2.

Alternative choices:

• Choice 1: $8^{15}$ is incorrect because it misapplies the exponent multiplication.
• Choice 3: $8^5$ is incorrect because it does not apply the power of a power rule.
• Choice 4: $8^2$ is incorrect and unrelated to the operation.

I am confident in the correctness of this solution.