Question

fully simplify using only positive exponents.\\(\frac{14x^{8}y^{6}}{21x^{8}y^{2}}\\)

Explanation:

Step1: Simplify the coefficients

Simplify \(\frac{14}{21}\) by dividing numerator and denominator by their greatest common divisor, which is 7.
\(\frac{14\div7}{21\div7}=\frac{2}{3}\)

Step2: Simplify the \(x\)-terms

Using the quotient rule for exponents \( \frac{a^m}{a^n}=a^{m - n} \), for \(x\)-terms: \(\frac{x^8}{x^8}=x^{8 - 8}=x^0 = 1\) (since any non - zero number to the power of 0 is 1)

Step3: Simplify the \(y\)-terms

Using the quotient rule for exponents \( \frac{a^m}{a^n}=a^{m - n} \), for \(y\)-terms: \(\frac{y^6}{y^2}=y^{6 - 2}=y^4\)

Step4: Multiply the simplified parts together

Multiply the simplified coefficient, \(x\)-term, and \(y\)-term: \(\frac{2}{3}\times1\times y^4=\frac{2y^4}{3}\)

Answer:

\(\frac{2y^{4}}{3}\)