Answer :
To solve the problem [tex]\((4x^2)^3\)[/tex], we need to apply the rules of exponents to find the equivalent expression. Here is a step-by-step explanation:
Step 1: Understand the expression
- The expression is [tex]\((4x^2)^3\)[/tex].
Step 2: Apply the power of a product rule
- According to the power of a product rule, [tex]\((a \cdot b)^n = a^n \cdot b^n\)[/tex]. This means we can separate the constants and variables and raise each to the power of 3.
Step 3: Apply the power rule for coefficients
- Take the constant part [tex]\(4\)[/tex]:
- The expression [tex]\(4^1\)[/tex] raised to the power 3 becomes [tex]\(4^3\)[/tex].
Step 4: Apply the power rule for variables
- Take the variable part [tex]\(x^2\)[/tex]:
- The expression [tex]\((x^2)\)[/tex] raised to the power 3 is [tex]\(x^{2 \times 3}\)[/tex], which is [tex]\(x^6\)[/tex].
Step 5: Evaluate the results
- Now, calculate [tex]\(4^3\)[/tex]. This is [tex]\(4 \times 4 \times 4 = 64\)[/tex].
- The variable part [tex]\(x^6\)[/tex] remains as is.
Step 6: Combine the results
- The equivalent expression [tex]\((4x^2)^3\)[/tex] becomes [tex]\(64x^6\)[/tex].
Therefore, the expression [tex]\((4x^2)^3\)[/tex] is equivalent to [tex]\(64x^6\)[/tex]. The correct choice from the given options is [tex]\(64x^6\)[/tex].
Step 1: Understand the expression
- The expression is [tex]\((4x^2)^3\)[/tex].
Step 2: Apply the power of a product rule
- According to the power of a product rule, [tex]\((a \cdot b)^n = a^n \cdot b^n\)[/tex]. This means we can separate the constants and variables and raise each to the power of 3.
Step 3: Apply the power rule for coefficients
- Take the constant part [tex]\(4\)[/tex]:
- The expression [tex]\(4^1\)[/tex] raised to the power 3 becomes [tex]\(4^3\)[/tex].
Step 4: Apply the power rule for variables
- Take the variable part [tex]\(x^2\)[/tex]:
- The expression [tex]\((x^2)\)[/tex] raised to the power 3 is [tex]\(x^{2 \times 3}\)[/tex], which is [tex]\(x^6\)[/tex].
Step 5: Evaluate the results
- Now, calculate [tex]\(4^3\)[/tex]. This is [tex]\(4 \times 4 \times 4 = 64\)[/tex].
- The variable part [tex]\(x^6\)[/tex] remains as is.
Step 6: Combine the results
- The equivalent expression [tex]\((4x^2)^3\)[/tex] becomes [tex]\(64x^6\)[/tex].
Therefore, the expression [tex]\((4x^2)^3\)[/tex] is equivalent to [tex]\(64x^6\)[/tex]. The correct choice from the given options is [tex]\(64x^6\)[/tex].
