Question

(\frac{(3y^{5})^{3}}{3y^{3}})

Explanation:

Step1: Simplify the numerator using power of a product rule

The power of a product rule states that \((ab)^n = a^n b^n\). For \((3y^5)^3\), we have \(3^3\times(y^5)^3\). Calculating \(3^3 = 27\) and using the power of a power rule \((a^m)^n=a^{mn}\), so \((y^5)^3 = y^{5\times3}=y^{15}\). So the numerator becomes \(27y^{15}\).
Now our expression is \(\frac{27y^{15}}{3y^3}\)

Step2: Simplify the coefficients and the variables separately

For the coefficients, we have \(\frac{27}{3}=9\). For the variables with exponents, we use the quotient rule of exponents \(\frac{a^m}{a^n}=a^{m - n}\), so \(\frac{y^{15}}{y^3}=y^{15 - 3}=y^{12}\)

Step3: Combine the results

Multiply the simplified coefficient and the simplified variable part together. So \(9\times y^{12}=9y^{12}\)

Answer:

\(9y^{12}\)