Question

express the following fraction in simplest form, only using positive exponents. \\(\frac{(3m^4t^5)^2}{2t^7}\\)

Explanation:

Step1: Apply power of a product rule

Using the rule \((ab)^n = a^n b^n\), we expand the numerator: \((3m^{4}t^{5})^{2}=3^{2}(m^{4})^{2}(t^{5})^{2}\). Calculating the powers, we get \(9m^{8}t^{10}\). So the fraction becomes \(\frac{9m^{8}t^{10}}{2t^{7}}\).

Step2: Apply quotient rule for exponents

For the variable \(t\), use the rule \(\frac{a^{m}}{a^{n}}=a^{m - n}\) (where \(a
eq0\), \(m>n\)). So \(\frac{t^{10}}{t^{7}}=t^{10 - 7}=t^{3}\). The \(m\) term and the constants remain as they are. Combining these, we have \(\frac{9m^{8}t^{3}}{2}\).

Answer:

\(\frac{9m^{8}t^{3}}{2}\)