Question

express the following fraction in simplest form, only using positive exponents.\\
\\(\frac{-3(x^2)^4}{-15x^7}\\)

Explanation:

Step1: Simplify the coefficient

Simplify the fraction of the coefficients \(\frac{-3}{-15}\).
\(\frac{-3}{-15}=\frac{3}{15}=\frac{1}{5}\)

Step2: Simplify the exponent of \(x\)

Use the power - of - a - power rule \((a^{m})^{n}=a^{mn}\) to simplify \((x^{2})^{4}\). We have \((x^{2})^{4}=x^{2\times4}=x^{8}\).
Now our expression becomes \(\frac{1}{5}\times\frac{x^{8}}{x^{7}}\).
Then use the quotient rule of exponents \(\frac{a^{m}}{a^{n}}=a^{m - n}\) (\(a
eq0\), \(m\) and \(n\) are real numbers) to simplify \(\frac{x^{8}}{x^{7}}\). Here, \(a = x\), \(m = 8\), \(n = 7\), so \(\frac{x^{8}}{x^{7}}=x^{8 - 7}=x\).

Step3: Combine the results

Multiply the simplified coefficient and the simplified \(x\) - term. \(\frac{1}{5}\times x=\frac{x}{5}\)

Answer:

\(\frac{x}{5}\)