To simplify the expression [tex]\((8x^2)^3\)[/tex], let's follow these steps:
1. Understand the Given Expression:
The given expression is [tex]\((8x^2)^3\)[/tex]. This means both the numerical coefficient (8) and the variable part ([tex]\(x^2\)[/tex]) are raised to the power of 3.
2. Apply the Power Rule:
The power rule [tex]\((a^m)^n = a^{mn}\)[/tex] tells us how to handle exponents in such scenarios.
- First, applying this rule to the coefficient:
[tex]\[
8^3
\][/tex]
Here, 8 is raised to the power of 3.
- Next, applying the same rule to the variable part:
[tex]\[
(x^2)^3
\][/tex]
According to the power rule, [tex]\((x^2)^3\)[/tex] equals [tex]\(x^{2 \cdot 3}\)[/tex].
3. Calculate the Exponents:
- For the coefficient:
[tex]\[
8^3 = 8 \times 8 \times 8 = 512
\][/tex]
- For the variable part:
[tex]\[
(x^2)^3 = x^{2 \cdot 3} = x^6
\][/tex]
4. Combine the Results:
Putting these together, the simplified form of the expression [tex]\((8x^2)^3\)[/tex] is:
[tex]\[
512x^6
\][/tex]
5. Choose the Correct Answer:
From the options provided:
- [tex]\(24x^6\)[/tex]
- [tex]\(512x^6\)[/tex]
- [tex]\(512x^5\)[/tex]
- [tex]\(24x^5\)[/tex]
The correct simplification is:
[tex]\[
512x^6
\][/tex]
Therefore, the correct answer is [tex]\( \boxed{512x^6} \)[/tex].
