Question

question
fully simplify using only positive exponents.
\\(\frac{6xy^5}{10x^6y^6}\\)

Explanation:

Step1: Simplify the coefficients

Simplify the fraction of the coefficients \(\frac{6}{10}\) by dividing numerator and denominator by their greatest common divisor, which is 2.
\(\frac{6\div2}{10\div2}=\frac{3}{5}\)

Step2: Simplify the \(x\)-terms

Use the quotient rule for exponents \( \frac{a^m}{a^n}=a^{m - n}\) for the \(x\)-terms. Here, \(m = 1\) and \(n = 6\) for \(x\) terms.
\(\frac{x}{x^{6}}=x^{1-6}=x^{-5}\). Since we need positive exponents, \(x^{-5}=\frac{1}{x^{5}}\)

Step3: Simplify the \(y\)-terms

Use the quotient rule for exponents for the \(y\)-terms. Here, \(m = 5\) and \(n = 6\) for \(y\) terms.
\(\frac{y^{5}}{y^{6}}=y^{5 - 6}=y^{-1}\). Since we need positive exponents, \(y^{-1}=\frac{1}{y}\)

Step4: Combine all the simplified terms

Multiply the simplified coefficient, \(x\)-term, and \(y\)-term together.
\(\frac{3}{5}\times\frac{1}{x^{5}}\times\frac{1}{y}=\frac{3}{5x^{5}y}\)

Answer:

\(\frac{3}{5x^{5}y}\)