The expression \( 4^{10} \) can be rewritten using the power rule of exponents. The power rule states that \( (a^m)^n = a^{m \cdot n} \).
To simplify \( 4^{10} \) using this rule, it can be expressed as:
\[ (4^2)^5 = 4^{2 \cdot 5} = 4^{10} \]
This means \( (4^2)^5 \) is equivalent to \( 4^{10} \). However, none of the given options directly show \( (4^2)^5 \).
Now, let's evaluate each of the provided options:
-
(4^2)^8 \[ (4^2)^8 = 4^{2 \cdot 8} = 4^{16} \] This is not equivalent to \( 4^{10} \).
-
(4^12)^2 \[ (4^{12})^2 = 4^{12 \cdot 2} = 4^{24} \] This is not equivalent to \( 4^{10} \).
-
(4^5)^2 \[ (4^5)^2 = 4^{5 \cdot 2} = 4^{10} \] This is equivalent to \( 4^{10} \).
-
4 . 4 . 4 . 4 . 4 . 4 . 4 This expression represents \( 4^7 \), which is also not equivalent to \( 4^{10} \).
Thus, the equivalent expanded expression for \( 4^{10} \) using the power rule of exponents from the options is:
(4^5)^2