Question

fully simplify using only positive exponents.
\frac{10x^{2}y^{3}}{4x^{5}y^{2}}

Explanation:

Step1: Simplify the coefficients

Simplify the fraction of the coefficients \( \frac{10}{4} \). We can divide both the numerator and the denominator by their greatest common divisor, which is 2. So, \( \frac{10\div2}{4\div2}=\frac{5}{2} \).

Step2: Simplify the \( x \)-terms

For the \( x \)-terms, we use the quotient rule of exponents \( \frac{a^m}{a^n}=a^{m - n} \). Here, \( a = x \), \( m = 2 \), and \( n = 5 \). So, \( \frac{x^{2}}{x^{5}}=x^{2 - 5}=x^{-3} \). But we need positive exponents, and by the definition of negative exponents, \( x^{-3}=\frac{1}{x^{3}} \).

Step3: Simplify the \( y \)-terms

For the \( y \)-terms, again use the quotient rule of exponents. Here, \( a = y \), \( m = 3 \), and \( n = 2 \). So, \( \frac{y^{3}}{y^{2}}=y^{3 - 2}=y^{1}=y \).

Step4: Combine the results

Now, combine the simplified coefficient, \( x \)-term, and \( y \)-term. We have \( \frac{5}{2}\times\frac{1}{x^{3}}\times y=\frac{5y}{2x^{3}} \).

Answer:

\( \frac{5y}{2x^{3}} \)