To divide the expression [tex]\(\frac{20 x^7 y^4 + 24 x y^2 + 12 x y^3}{4 x y^3}\)[/tex], we will simplify each term in the numerator by the denominator [tex]\(4 x y^3\)[/tex].
Let's break it down step-by-step:
1. First Term: [tex]\( \frac{20 x^7 y^4}{4 x y^3} \)[/tex]
- Dividing the coefficients: [tex]\(\frac{20}{4} = 5\)[/tex]
- Dividing the [tex]\(x\)[/tex] terms: [tex]\(x^7 / x = x^{7-1} = x^6\)[/tex]
- Dividing the [tex]\(y\)[/tex] terms: [tex]\(y^4 / y^3 = y^{4-3} = y\)[/tex]
So, the simplified form of the first term is [tex]\(5 x^6 y\)[/tex].
2. Second Term: [tex]\( \frac{24 x y^2}{4 x y^3} \)[/tex]
- Dividing the coefficients: [tex]\(\frac{24}{4} = 6\)[/tex]
- Dividing the [tex]\(x\)[/tex] terms: [tex]\(x / x = 1\)[/tex]
- Dividing the [tex]\(y\)[/tex] terms: [tex]\(y^2 / y^3 = y^{2-3} = y^{-1} = \frac{1}{y}\)[/tex]
So, the simplified form of the second term is [tex]\(6 \frac{1}{y} = \frac{6}{y}\)[/tex].
3. Third Term: [tex]\( \frac{12 x y^3}{4 x y^3} \)[/tex]
- Dividing the coefficients: [tex]\(\frac{12}{4} = 3\)[/tex]
- Dividing the [tex]\(x\)[/tex] terms: [tex]\(x / x = 1\)[/tex]
- Dividing the [tex]\(y\)[/tex] terms: [tex]\(y^3 / y^3 = 1\)[/tex]
So, the simplified form of the third term is [tex]\(3\)[/tex].
After simplifying each term, we combine the results:
[tex]\[
5 x^6 y + \frac{6}{y} + 3
\][/tex]
Thus, the simplified expression is:
[tex]\[
\boxed{5 x^6 y + \frac{6}{y} + 3}
\][/tex]
