To simplify the expression [tex]\((4g^{10})^4\)[/tex], we need to use the rules of exponents. The essential rules we will use are:
1. [tex]\((a^m)^n = a^{m \cdot n}\)[/tex]
2. [tex]\((a \cdot b)^n = a^n \cdot b^n\)[/tex]
Here's the step-by-step process:
1. Identify the Base and Exponent:
The given expression is [tex]\((4g^{10})^4\)[/tex]. Here, 4 is the coefficient (base for the numerical part), and [tex]\(g^{10}\)[/tex] is the base with exponent 10 for the variable part.
2. Apply the Power to Both Terms:
Using the exponent rule [tex]\((a \cdot b)^n = a^n \cdot b^n\)[/tex], we can split this as:
[tex]\[
(4g^{10})^4 = 4^4 \cdot (g^{10})^4
\][/tex]
3. Simplify the Numerical Part:
Calculate [tex]\(4^4\)[/tex]:
[tex]\[
4^4 = 4 \cdot 4 \cdot 4 \cdot 4 = 256
\][/tex]
4. Simplify the Variable Part:
For [tex]\((g^{10})^4\)[/tex], use the exponent rule [tex]\((a^m)^n = a^{m \cdot n}\)[/tex]:
[tex]\[
(g^{10})^4 = g^{10 \cdot 4} = g^{40}
\][/tex]
5. Combine the Results:
Now combine the simplified numerical part and the simplified variable part:
[tex]\[
(4g^{10})^4 = 256g^{40}
\][/tex]
Therefore, the simplified expression is:
[tex]\[
\boxed{256g^{40}}
\][/tex]
Among the given options, [tex]\(256g^{40}\)[/tex] is the correct simplified form.
