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Choose the correct simplification of the expression [tex] \left(8 x^2\right)^3 [/tex].
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].
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