Question

question
fully simplify using only positive exponents.
\\(\frac{30x^{6}y^{8}}{12x^{7}y^{5}}\\)

Explanation:

Step1: Simplify the coefficients

Simplify the fraction of the coefficients 30 and 12. We find the greatest common divisor (GCD) of 30 and 12, which is 6. Then divide both numerator and denominator by 6:
$\frac{30\div6}{12\div6}=\frac{5}{2}$

Step2: Simplify the $x$ terms

Use the quotient rule for exponents, $a^m\div a^n = a^{m - n}$. For the $x$ terms, we have $x^6\div x^7$:
$x^{6 - 7}=x^{-1}$
Since we need positive exponents, we know that $x^{-1}=\frac{1}{x}$

Step3: Simplify the $y$ terms

Again, use the quotient rule for exponents for the $y$ terms, $y^8\div y^5$:
$y^{8 - 5}=y^{3}$

Step4: Combine all the simplified terms

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

Answer:

$\frac{5y^{3}}{2x}$