If the expression 410 was simplified using the Power Rule of Exponents, what is an equivalent expanded expression? (4^12)^2 4⋅4⋅4⋅4⋅4⋅4⋅4 4 (4^2)^8

Question

(4^5)^2

Answer 1

To simplify the expression \( (4^{12})^2 \) using the Power Rule of Exponents, we can apply the rule that states \( (a^m)^n = a^{m \cdot n} \).

Applying this rule to the expression:

\[ (4^{12})^2 = 4^{12 \cdot 2} = 4^{24} \]

Next, we can expand \( 4^{24} \) as:

\[ 4^{24} = 4 \cdot 4 \cdot 4 \cdots \text{(24 times)} \]

Now, let's check the other expressions provided in the question to find an equivalent expanded form to \( 4^{24} \):

1. \( 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \) – This does not specify how many times 4 is multiplied, so it's not clear from this alone.

2. \( (4^2)^8 \) – Using the power rule again: \[ (4^2)^8 = 4^{2 \cdot 8} = 4^{16} \] This is not equivalent to \( 4^{24} \).

3. \( (4^5)^2 \) – Using the power rule: \[ (4^5)^2 = 4^{5 \cdot 2} = 4^{10} \] This is also not equivalent to \( 4^{24} \).

The only expression that aligns with \( 4^{24} \) is:

\[ 4 \cdot 4 \cdots \text{(24 times)} \]

So, the final answer for the expanded equivalent of \( (4^{12})^2 \) is \( 4^{24} \) or \( 4 \cdot 4 \cdot \ldots \) (24 times).