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 the power of a product rule on the denominator's exponent

First, simplify the coefficient \(\frac{10}{5} = 2\). Then, use the power of a product rule \((ab)^n=a^nb^n\) and the power of a power rule \((a^m)^n = a^{mn}\) on \((k^{-5}b^{2})^{-3}\):
\((k^{-5}b^{2})^{-3}=k^{(-5)\times(-3)}b^{2\times(-3)}=k^{15}b^{-6}\)
So the expression becomes \(2\times\frac{k^{-10}b^{-9}}{k^{15}b^{-6}}\)

Step2: Use the quotient rule for exponents \( \frac{a^m}{a^n}=a^{m - n}\)

For the \(k\) terms: \(\frac{k^{-10}}{k^{15}}=k^{-10-15}=k^{-25}=\frac{1}{k^{25}}\) (since \(a^{-n}=\frac{1}{a^n}\))
For the \(b\) terms: \(\frac{b^{-9}}{b^{-6}}=b^{-9 - (-6)}=b^{-9 + 6}=b^{-3}=\frac{1}{b^{3}}\)

Step3: Combine the terms

Multiply the coefficient with the simplified \(k\) and \(b\) terms:
\(2\times\frac{1}{k^{25}}\times\frac{1}{b^{3}}=\frac{2}{k^{25}b^{3}}\)

Answer:

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