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

A. [tex] 24 x^6 [/tex]

B. [tex] 512 x^6 [/tex]

C. [tex] 512 x^5 [/tex]

D. [tex] 24 x^5 [/tex]



Answer :

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].