Simplify:

[tex]\[
\frac{\left(2 y^2\right)^4}{\left(5 y^6\right)^2}
\][/tex]

Write your answer using only positive exponents.



Answer :

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]