Answer :
To simplify the expression [tex]\((4x^2)^3\)[/tex], we can use the rules of exponents. Here's a step-by-step solution:
1. Identify the Components: The expression inside the parentheses is [tex]\(4x^2\)[/tex], and it is being raised to the power of 3.
2. Apply the Power Rule: According to the power rule for exponents [tex]\((a^m)^n = a^{m \cdot n}\)[/tex], we need to apply the exponent to both the coefficient and the variable inside the parentheses.
3. Simplify the Coefficient:
- Start with the coefficient 4.
- Raise 4 to the power of 3: [tex]\(4^3 = 4 \times 4 \times 4 = 64\)[/tex].
4. Simplify the Variable:
- The variable part is [tex]\(x^2\)[/tex].
- Apply the power of 3 to [tex]\(x^2\)[/tex]: [tex]\((x^2)^3\)[/tex].
- Using the power rule again, [tex]\((x^2)^3 = x^{2 \times 3} = x^6\)[/tex].
5. Combine the Results:
- Combine the simplified coefficient and the variable: [tex]\(64x^6\)[/tex].
Therefore, the expression [tex]\((4x^2)^3\)[/tex] simplifies to [tex]\(64x^6\)[/tex].
So, the equivalent expression is [tex]\(64x^6\)[/tex], which matches the second option in the list of choices.
1. Identify the Components: The expression inside the parentheses is [tex]\(4x^2\)[/tex], and it is being raised to the power of 3.
2. Apply the Power Rule: According to the power rule for exponents [tex]\((a^m)^n = a^{m \cdot n}\)[/tex], we need to apply the exponent to both the coefficient and the variable inside the parentheses.
3. Simplify the Coefficient:
- Start with the coefficient 4.
- Raise 4 to the power of 3: [tex]\(4^3 = 4 \times 4 \times 4 = 64\)[/tex].
4. Simplify the Variable:
- The variable part is [tex]\(x^2\)[/tex].
- Apply the power of 3 to [tex]\(x^2\)[/tex]: [tex]\((x^2)^3\)[/tex].
- Using the power rule again, [tex]\((x^2)^3 = x^{2 \times 3} = x^6\)[/tex].
5. Combine the Results:
- Combine the simplified coefficient and the variable: [tex]\(64x^6\)[/tex].
Therefore, the expression [tex]\((4x^2)^3\)[/tex] simplifies to [tex]\(64x^6\)[/tex].
So, the equivalent expression is [tex]\(64x^6\)[/tex], which matches the second option in the list of choices.
