Question

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
(4^5)^2

Answer 1

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

Let's simplify each of the provided options using the Power Rule and identify the equivalent expanded expression.

1. \((4^{12})^2\):
\[
(4^{12})^2 = 4^{12 \cdot 2} = 4^{24}
\]

2. \(4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \):
This is the same as \(4^7\) because there are seven factors of 4.

3. \((4^2)^8\):
\[
(4^2)^8 = 4^{2 \cdot 8} = 4^{16}
\]

4. \((4^5)^2\):
\[
(4^5)^2 = 4^{5 \cdot 2} = 4^{10}
\]

Now we have the following equivalent expressions:
- \((4^{12})^2 = 4^{24}\)
- \(4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 = 4^{7}\)
- \((4^2)^8 = 4^{16}\)
- \((4^5)^2 = 4^{10}\)

None of these expressions are equivalent to 410 when simplified, as \(410\) is not an expression where a Power Rule is applicable.

However, if you are asking which on the list represents a power of 4 in "expanded" form, it seems none of these options simplifies directly to \(4^{410}\) or any interpretation of \(410\) requested. If you have further context or a specific form of \(410\) in mind, please provide that for clearer assistance!