Question

fully simplify using only positive exponents.\\(\frac{24x^{3}y^{8}}{8x^{4}y^{6}}\\)

Explanation:

Step1: Simplify the coefficient

Divide the coefficient 24 by 8.
$\frac{24}{8} = 3$

Step2: Simplify the \(x\)-terms

Use the quotient rule for exponents \( \frac{a^m}{a^n}=a^{m - n} \) for \(x\)-terms. Here, \(m = 3\) and \(n = 4\), so \( \frac{x^3}{x^4}=x^{3 - 4}=x^{-1}\). Since we need positive exponents, \(x^{-1}=\frac{1}{x}\).

Step3: Simplify the \(y\)-terms

Use the quotient rule for exponents for \(y\)-terms. Here, \(m = 8\) and \(n = 6\), so \( \frac{y^8}{y^6}=y^{8 - 6}=y^{2}\).

Step4: Combine the results

Multiply the simplified coefficient, \(x\)-term, and \(y\)-term together. So we have \(3\times\frac{1}{x}\times y^{2}=\frac{3y^{2}}{x}\).

Answer:

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