Evaluate the Nested Expression: (2²)³ Using Exponent Rules

We are given the expression $(2^2)^3$ and need to simplify it using the laws of exponents and identify the corresponding expression among the choices.

To simplify the expression $(2^2)^3$, we use the "power of a power" rule, which states that $(a^m)^n = a^{m \times n}$.

Applying this rule to our expression, we have:

$(2^2)^3 = 2^{2 \times 3}$

Calculating the new exponent:

$2 \times 3 = 6$

Thus, the expression simplifies to:

$2^6$

Now, let's compare our result $2^6$ with the given choices:

• Choice 1: $2^{2+3} = 2^5$ - Incorrect, as our expression evaluates to $2^6$, not $2^5$.
• Choice 2: $2^{2-3} = 2^{-1}$ - Incorrect, as our expression evaluates to $2^6$, not $2^{-1}$.
• Choice 3: $2^{\frac{2}{3}}$ - Incorrect, as our expression evaluates to $2^6$, not a fractional exponent expression.
• Choice 4: $2^{2 \times 3} = 2^6$ - Correct, as this matches our simplified expression.

Therefore, the correct choice is Choice 4: $2^{2 \times 3}$.