Sure, let's analyze the given problem step-by-step to determine which of the given options is equivalent to the expression [tex]\(\left(-7 x^3\right)^2\)[/tex].
Given:
[tex]\[
\left(-7 x^3\right)^2
\][/tex]
Step 1: Understand the properties of exponents.
Recall that when you raise a product to a power, you raise each factor in the product to that power. More specifically:
[tex]\[
(a \cdot b)^n = a^n \cdot b^n
\][/tex]
and additionally, raising a power to another power means you multiply the exponents:
[tex]\[
(x^a)^b = x^{a \cdot b}
\][/tex]
Step 2: Apply these properties to the given expression.
Start with the expression:
[tex]\[
\left(-7 x^3\right)^2
\][/tex]
Step 3: Raise each factor inside the parentheses to the power of 2.
[tex]\[
(-7)^2 \cdot (x^3)^2
\][/tex]
Step 4: Calculate the powers separately.
[tex]\[
(-7)^2 = 49
\][/tex]
[tex]\[
(x^3)^2 = x^{3 \cdot 2} = x^6
\][/tex]
Step 5: Combine the results.
[tex]\[
49 \cdot x^6 = 49 x^6
\][/tex]
Therefore, the expression [tex]\(\left(-7 x^3\right)^2\)[/tex] simplifies to:
[tex]\[
49 x^6
\][/tex]
So the correct answer is:
(3) [tex]\(49 x^6\)[/tex]
