Question

fully simplify using only positive exponents. \\(\frac{3x^2y^3}{9x^8y^6}\\)

Explanation:

Step1: Simplify the coefficient

Simplify the fraction of the coefficients: $\frac{3}{9}=\frac{1}{3}$.

Step2: Simplify the \(x\)-terms

Use the exponent rule $\frac{a^m}{a^n}=a^{m - n}$ for \(x\)-terms: $\frac{x^2}{x^8}=x^{2 - 8}=x^{-6}=\frac{1}{x^6}$ (since we need positive exponents, we apply $a^{-n}=\frac{1}{a^n}$).

Step3: Simplify the \(y\)-terms

Use the exponent rule for \(y\)-terms: $\frac{y^3}{y^6}=y^{3 - 6}=y^{-3}=\frac{1}{y^3}$.

Step4: Combine all simplified terms

Multiply the simplified coefficient, \(x\)-term, and \(y\)-term together: $\frac{1}{3}\times\frac{1}{x^6}\times\frac{1}{y^3}=\frac{1}{3x^6y^3}$.

Answer:

$\frac{1}{3x^6y^3}$