To solve the equation [tex]\(\left(x^2\right)^y = x^{16}\)[/tex], we need to use properties of exponents. Let's go through the steps in detail:
1. Apply the power of a power property to the left side of the equation:
The property [tex]\( (a^m)^n = a^{m \cdot n} \)[/tex] allows us to rewrite [tex]\(\left(x^2\right)^y\)[/tex]:
[tex]\[
\left(x^2\right)^y = x^{2 \cdot y}
\][/tex]
2. Rewrite the equation using the exponent property:
Substituting the left side of the original equation, we get:
[tex]\[
x^{2y} = x^{16}
\][/tex]
3. Since the bases are the same, set the exponents equal to each other:
For the equation [tex]\( x^{a} = x^{b} \)[/tex] to be true, [tex]\(a\)[/tex] must equal [tex]\(b\)[/tex]. Therefore, we have:
[tex]\[
2y = 16
\][/tex]
4. Solve for [tex]\(y\)[/tex]:
Isolate [tex]\(y\)[/tex] by dividing both sides of the equation by 2:
[tex]\[
y = \frac{16}{2}
\][/tex]
[tex]\[
y = 8
\][/tex]
Thus, the value of [tex]\(y\)[/tex] is [tex]\(8\)[/tex]. Therefore, the answer is:
[tex]\[
\boxed{8}
\][/tex]
