To solve the equation [tex]\(\left(x^2\right)^y = x^{16}\)[/tex], we need to use the properties of exponents. Let's proceed step-by-step.
1. Rewrite the left-hand side using the exponentiation rule: [tex]\((a^m)^n = a^{m \cdot n}\)[/tex]. Therefore, [tex]\(\left(x^2\right)^y\)[/tex] becomes:
[tex]\[
(x^2)^y = x^{2y}
\][/tex]
2. Equate the exponents on both sides of the equation since the bases on both sides are the same (both are [tex]\(x\)[/tex]):
[tex]\[
x^{2y} = x^{16}
\][/tex]
3. Solve for [tex]\(y\)[/tex] by setting the exponents equal to each other:
[tex]\[
2y = 16
\][/tex]
4. Divide both sides by 2:
[tex]\[
y = \frac{16}{2}
\][/tex]
5. Simplify the right-hand side:
[tex]\[
y = 8
\][/tex]
So, [tex]\(y = 8\)[/tex].
Given the choices:
A) 4
B) 6
C) 8
D) 14
The correct equivalent value of [tex]\(y\)[/tex] is [tex]\(8\)[/tex], which corresponds to option C.
