Question

fully simplify using only positive exponents.\\(\frac{27x^{3}y^{5}}{3x^{4}y^{2}}\\)

Explanation:

Step1: Simplify the coefficient

Divide the coefficient 27 by 3.
$\frac{27}{3} = 9$

Step2: Simplify the \(x\)-terms

Use the quotient rule for exponents \( \frac{a^m}{a^n}=a^{m - n} \) for \(x\)-terms.
$\frac{x^3}{x^4}=x^{3 - 4}=x^{-1}=\frac{1}{x}$ (but we want positive exponents, so we'll handle this later in the combined form)

Step3: Simplify the \(y\)-terms

Use the quotient rule for exponents for \(y\)-terms.
$\frac{y^5}{y^2}=y^{5 - 2}=y^{3}$

Step4: Combine all terms

Multiply the simplified coefficient, \(x\)-term, and \(y\)-term.
$9\times\frac{1}{x}\times y^{3}=\frac{9y^{3}}{x}$

Answer:

$\frac{9y^{3}}{x}$