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 \( \frac{10}{4} \) by dividing both numerator and denominator by their greatest common divisor, which is 2.
\( \frac{10\div2}{4\div2}=\frac{5}{2} \)

Step2: Simplify the \( x \)-terms using exponent rules

For the \( x \)-terms \( \frac{x^{2}}{x^{5}} \), use the rule of exponents \( \frac{a^{m}}{a^{n}} = a^{m - n} \) (where \( a
eq0 \), \( m \) and \( n \) are real numbers). Here, \( m = 2 \) and \( n = 5 \), so:
\( x^{2-5}=x^{-3} \)
Since we need only positive exponents, recall that \( a^{-n}=\frac{1}{a^{n}} \), so \( x^{-3}=\frac{1}{x^{3}} \)

Step3: Simplify the \( y \)-terms using exponent rules

For the \( y \)-terms \( \frac{y^{3}}{y^{2}} \), use the same exponent rule \( \frac{a^{m}}{a^{n}}=a^{m - n} \). Here, \( m = 3 \) and \( n = 2 \), so:
\( y^{3 - 2}=y^{1}=y \)

Step4: Combine all the simplified terms

Multiply the simplified coefficient, \( x \)-term, and \( y \)-term together:
\( \frac{5}{2}\times\frac{1}{x^{3}}\times y=\frac{5y}{2x^{3}} \)

Answer:

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