Question

simplify.
\\(\frac{(2y^{3})^{4}}{(8y^{2})^{2}}\\)
write your answer using only positive exponents.

Explanation:

Step1: Apply power of a product rule

For \((2y^3)^4\), we have \(2^4\times(y^3)^4 = 16y^{12}\) (using \((ab)^n=a^n b^n\) and \((a^m)^n=a^{mn}\)). For \((8y^2)^2\), we have \(8^2\times(y^2)^2 = 64y^4\) (using the same rules). So the expression becomes \(\frac{16y^{12}}{64y^4}\).

Step2: Simplify the coefficients and exponents

Simplify the coefficient: \(\frac{16}{64}=\frac{1}{4}\). For the exponents of \(y\), use the rule \(\frac{a^m}{a^n}=a^{m - n}\), so \(y^{12-4}=y^8\). Combining these, we get \(\frac{1}{4}y^8\) or \(\frac{y^8}{4}\).

Answer:

\(\frac{y^8}{4}\)