To simplify the given expression:
[tex]\[
\frac{\left(2 y^2\right)^4}{\left(5 y^6\right)^2}
\][/tex]
we will follow these steps:
1. Simplify the numerator:
[tex]\[
(2y^2)^4
\][/tex]
When an expression is raised to a power, we need to raise both the coefficient and the variable to the power separately. Thus:
[tex]\[
(2y^2)^4 = 2^4 \cdot (y^2)^4
\][/tex]
Now, compute the powers:
[tex]\[
2^4 = 16 \quad \text{and} \quad (y^2)^4 = y^{2 \cdot 4} = y^8
\][/tex]
Therefore, the numerator simplifies to:
[tex]\[
16 y^8
\][/tex]
2. Simplify the denominator:
[tex]\[
(5y^6)^2
\][/tex]
As with the numerator, raise both the coefficient and the variable to the power separately. Thus:
[tex]\[
(5y^6)^2 = 5^2 \cdot (y^6)^2
\][/tex]
Now, compute the powers:
[tex]\[
5^2 = 25 \quad \text{and} \quad (y^6)^2 = y^{6 \cdot 2} = y^{12}
\][/tex]
Therefore, the denominator simplifies to:
[tex]\[
25 y^{12}
\][/tex]
3. Combine the simplified numerator and denominator:
[tex]\[
\frac{16 y^8}{25 y^{12}}
\][/tex]
4. Simplify the fraction by dividing the terms:
To simplify the fraction, we divide the coefficients and apply the properties of exponents to the variables.
[tex]\[
\frac{16}{25} \cdot \frac{y^8}{y^{12}}
\][/tex]
Since [tex]\(y^8\)[/tex] and [tex]\(y^{12}\)[/tex] have the same base, subtract the exponents:
[tex]\[
y^{8 - 12} = y^{-4}
\][/tex]
Therefore, the fraction simplifies to:
[tex]\[
\frac{16}{25} \cdot y^{-4}
\][/tex]
5. Express the result with only positive exponents:
Recall that [tex]\(y^{-4}\)[/tex] can be written as [tex]\(\frac{1}{y^4}\)[/tex]:
[tex]\[
\frac{16}{25} \cdot \frac{1}{y^4} = \frac{16}{25 y^4}
\][/tex]
Thus, the simplified form of the given expression is:
[tex]\[
\boxed{\frac{16}{25 y^4}}
\][/tex]
