Question

express the following fraction in simplest form using only positive exponents.\\(\frac{(3u^2)^5}{3u^4}\\)

Explanation:

Step1: Apply the power of a product rule to the numerator

The power of a product rule states that \((ab)^n = a^n b^n\). So for \((3u^2)^5\), we have \(3^5(u^2)^5\). Calculating \(3^5 = 243\) and using the power of a power rule \((a^m)^n=a^{mn}\), we get \((u^2)^5 = u^{2\times5}=u^{10}\). So the numerator becomes \(243u^{10}\).
Now our expression is \(\frac{243u^{10}}{3u^4}\).

Step2: Simplify the coefficients and the variables separately

For the coefficients, we divide \(243\) by \(3\). \(\frac{243}{3}=81\).
For the variables with exponents, we use the quotient of powers rule \( \frac{a^m}{a^n}=a^{m - n}\) (where \(a
eq0\) and \(m>n\)). So for \(u\), we have \(u^{10-4}=u^6\).

Step3: Combine the simplified coefficient and variable

Multiplying the simplified coefficient and the simplified variable together, we get \(81u^6\).

Answer:

\(81u^6\)