Question

express the following fraction in simplest form, only using positive exponents.\\(\frac{10k^{-10}b^{-9}}{5(k^{-5}b^2)^{-3}}\\)

Explanation:

Step1: Simplify the coefficient and use exponent rule \((a^m)^n = a^{mn}\)

First, simplify the coefficient \(\frac{10}{5}=2\). Then, for the denominator's exponent part \((k^{-5}b^{2})^{-3}\), apply the power - of - a - product rule: \((k^{-5})^{-3}(b^{2})^{-3}\). Using the exponent rule \((a^m)^n=a^{mn}\), we get \(k^{(-5)\times(-3)}b^{2\times(-3)} = k^{15}b^{-6}\). So the original expression becomes \(\frac{10k^{-10}b^{-9}}{5k^{15}b^{-6}}\) (after substituting the simplified denominator exponent part).

Step2: Simplify the fraction of coefficients and use exponent rule \(\frac{a^m}{a^n}=a^{m - n}\) for like bases

The coefficient fraction \(\frac{10}{5} = 2\). For the \(k\) terms: \(\frac{k^{-10}}{k^{15}}=k^{-10 - 15}=k^{-25}\). For the \(b\) terms: \(\frac{b^{-9}}{b^{-6}}=b^{-9-(-6)}=b^{-9 + 6}=b^{-3}\). Now our expression is \(2k^{-25}b^{-3}\).

Step3: Convert negative exponents to positive exponents

Using the rule \(a^{-n}=\frac{1}{a^{n}}\), we can rewrite \(k^{-25}=\frac{1}{k^{25}}\) and \(b^{-3}=\frac{1}{b^{3}}\). So the expression becomes \(\frac{2}{k^{25}b^{3}}\).

Answer:

\(\frac{2}{k^{25}b^{3}}\)