Question

4.) \frac{-4x^{5}y^{8}}{16x^{5}y^{6}}

Explanation:

Step1: Simplify the coefficients

Simplify the fraction of the coefficients \(\frac{-4}{16}\). We can divide both the numerator and the denominator by their greatest common divisor, which is 4. So \(\frac{-4\div4}{16\div4}=\frac{-1}{4}\).

Step2: Simplify the \(x\)-terms

For the \(x\)-terms, we use the rule of exponents \(\frac{a^m}{a^n}=a^{m - n}\). Here, \(m = 5\) and \(n = 5\), so \(\frac{x^{5}}{x^{5}}=x^{5 - 5}=x^{0}=1\) (since any non - zero number to the power of 0 is 1).

Step3: Simplify the \(y\)-terms

For the \(y\)-terms, using the same exponent rule \(\frac{a^m}{a^n}=a^{m - n}\), with \(m = 8\) and \(n = 6\), we have \(\frac{y^{8}}{y^{6}}=y^{8 - 6}=y^{2}\).

Step4: Combine the results

Multiply the simplified coefficient, \(x\)-term, and \(y\)-term together. We have \(\frac{-1}{4}\times1\times y^{2}=-\frac{y^{2}}{4}\).

Answer:

\(-\frac{y^{2}}{4}\)