Question

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

Explanation:

Step1: Simplify the denominator using power of a product rule

The power of a product rule states that \((ab)^n = a^n b^n\). So, for \((2t^5)^2\), we have \(2^2\times(t^5)^2\). Calculating \(2^2 = 4\) and using the power of a power rule \((a^m)^n=a^{mn}\), we get \((t^5)^2 = t^{5\times2}=t^{10}\). So the denominator becomes \(4t^{10}\). Now the fraction is \(\frac{3t^2}{4t^{10}}\).

Step2: Simplify the variable part using quotient rule for exponents

The quotient rule for exponents is \(\frac{a^m}{a^n}=a^{m - n}\) (when \(a
eq0\)). For the variable \(t\), we have \(\frac{t^2}{t^{10}}=t^{2 - 10}=t^{-8}\). But we need only positive exponents, and \(a^{-n}=\frac{1}{a^n}\), so \(t^{-8}=\frac{1}{t^8}\).

Step3: Combine the constant and variable parts

Now we have \(\frac{3}{4}\times\frac{1}{t^8}=\frac{3}{4t^8}\).

Answer:

\(\frac{3}{4t^{8}}\)