Answer :
Sure! Let's simplify the expression [tex]\(\left(4^2 x^3\right)^3\)[/tex] step-by-step.
1. Simplify inside the parentheses:
- We have [tex]\(4^2\)[/tex], which means 4 multiplied by itself: [tex]\(4 \times 4 = 16\)[/tex].
- So, the expression inside the parentheses becomes [tex]\(16x^3\)[/tex].
2. Raise the entire expression to the third power:
- Now we take [tex]\((16x^3)^3\)[/tex].
3. Apply the power of a product and power of a power rule:
- Using the rule [tex]\((a \cdot b)^n = a^n \cdot b^n\)[/tex], we get [tex]\((16)^3 \cdot (x^3)^3\)[/tex].
4. Calculate [tex]\(16^3\)[/tex]:
- [tex]\(16 \times 16 = 256\)[/tex].
- [tex]\(256 \times 16 = 4096\)[/tex].
- So, [tex]\(16^3 = 4096\)[/tex].
5. Use the power of a power rule for [tex]\(x^3\)[/tex]:
- The rule [tex]\((x^m)^n = x^{m \cdot n}\)[/tex] applies here.
- [tex]\((x^3)^3 = x^{3 \cdot 3} = x^9\)[/tex].
6. Combine the results:
- The expression [tex]\((16x^3)^3\)[/tex] simplifies to [tex]\(4096x^9\)[/tex].
Therefore, the simplified form is [tex]\(4096x^9\)[/tex], which matches option B.
1. Simplify inside the parentheses:
- We have [tex]\(4^2\)[/tex], which means 4 multiplied by itself: [tex]\(4 \times 4 = 16\)[/tex].
- So, the expression inside the parentheses becomes [tex]\(16x^3\)[/tex].
2. Raise the entire expression to the third power:
- Now we take [tex]\((16x^3)^3\)[/tex].
3. Apply the power of a product and power of a power rule:
- Using the rule [tex]\((a \cdot b)^n = a^n \cdot b^n\)[/tex], we get [tex]\((16)^3 \cdot (x^3)^3\)[/tex].
4. Calculate [tex]\(16^3\)[/tex]:
- [tex]\(16 \times 16 = 256\)[/tex].
- [tex]\(256 \times 16 = 4096\)[/tex].
- So, [tex]\(16^3 = 4096\)[/tex].
5. Use the power of a power rule for [tex]\(x^3\)[/tex]:
- The rule [tex]\((x^m)^n = x^{m \cdot n}\)[/tex] applies here.
- [tex]\((x^3)^3 = x^{3 \cdot 3} = x^9\)[/tex].
6. Combine the results:
- The expression [tex]\((16x^3)^3\)[/tex] simplifies to [tex]\(4096x^9\)[/tex].
Therefore, the simplified form is [tex]\(4096x^9\)[/tex], which matches option B.
