Evaluate (10³)³: Solving Nested Powers of 10

To solve this problem, we will proceed with the following steps:

• Identify the expression structure.
• Apply the power of a power rule for exponents.
• Simplify the expression.

Now, let's work through each step in detail:

Step 1: Identify the expression structure.
We have the expression $(10^3)^3$. This indicates a power of a power where the base is 10, the inner exponent is 3, and the entire expression is raised to another power of 3.

Step 2: Apply the power of a power rule.
The rule states $(a^m)^n = a^{m \times n}$. Applying this to our specific expression gives us:

$\left(10^3\right)^3 = 10^{3 \times 3}$

Step 3: Perform the multiplication in the exponent.
Calculating $3 \times 3$, we get $9$. Thus, the expression simplifies to:

$10^9$

Therefore, the solution to the problem is:

$\boxed{10^{3 \times 3}}$

Examining the provided choices:

• Choice 1: $10^{3+3}$ - Incorrect, because it uses addition instead of multiplication.
• Choice 2: $10^{3 \times 3}$ - Correct, as it matches our derived expression.
• Choice 3: $10^{\frac{3}{3}}$ - Incorrect, because it uses division instead of multiplication.
• Choice 4: $10^{3-3}$ - Incorrect, because it uses subtraction instead of multiplication.

The correct answer is $10^{3 \times 3}$, which is represented by Choice 2.