Question

question
express the following fraction in simplest form, only using positive exponents.
\\(\frac{-2(x^2p^3)^{-1}}{3x^{10}p^{-1}}\\)

Explanation:

Step1: Apply exponent rule \((ab)^n = a^n b^n\) and \((a^m)^n = a^{mn}\)

First, simplify the numerator \(-2(x^{2}p^{3})^{-1}\). Using \((a^m)^n = a^{mn}\), we get \(-2x^{2\times(-1)}p^{3\times(-1)}=-2x^{-2}p^{-3}\). So the fraction becomes \(\frac{-2x^{-2}p^{-3}}{3x^{10}p^{-1}}\).

Step2: Apply exponent rule \(\frac{a^m}{a^n}=a^{m - n}\)

For the \(x\)-terms: \(\frac{x^{-2}}{x^{10}}=x^{-2 - 10}=x^{-12}=\frac{1}{x^{12}}\) (since \(a^{-n}=\frac{1}{a^n}\)). For the \(p\)-terms: \(\frac{p^{-3}}{p^{-1}}=p^{-3-(-1)}=p^{-2}=\frac{1}{p^{2}}\). The constant terms remain \(\frac{-2}{3}\).

Step3: Combine all terms

Multiply the constant, \(x\)-term, and \(p\)-term together: \(\frac{-2}{3}\times\frac{1}{x^{12}}\times\frac{1}{p^{2}}=\frac{-2}{3x^{12}p^{2}}\).

Answer:

\(\boldsymbol{\frac{-2}{3x^{12}p^{2}}}\)