Question

fully simplify using only positive exponents. \\(\frac{10x^{6}y^{7}}{4x^{7}y^{2}}\\)

Explanation:

Step1: Simplify the coefficients

Simplify the fraction of the coefficients \(\frac{10}{4}\). We can divide both the numerator and the denominator by their greatest common divisor, which is 2. So \(\frac{10\div2}{4\div2}=\frac{5}{2}\).

Step2: Simplify the \(x\)-terms

Using the quotient rule for exponents \( \frac{a^m}{a^n}=a^{m - n}\), for the \(x\)-terms \(\frac{x^{6}}{x^{7}}\), we have \(x^{6-7}=x^{-1}\). But we need positive exponents, and \(x^{-1}=\frac{1}{x}\).

Step3: Simplify the \(y\)-terms

Using the quotient rule for exponents again, for the \(y\)-terms \(\frac{y^{7}}{y^{2}}\), we get \(y^{7 - 2}=y^{5}\).

Step4: Combine all the simplified terms

Now we combine the simplified coefficient, \(x\)-term, and \(y\)-term. We have \(\frac{5}{2}\times\frac{1}{x}\times y^{5}=\frac{5y^{5}}{2x}\).

Answer:

\(\frac{5y^{5}}{2x}\)