To solve the given expression [tex]\(4 x^{\frac{2}{3}} y^4 \sqrt[3]{x y^3}\)[/tex] and write it in the form [tex]\(a x^b y^c\)[/tex], we need to follow several steps. Here is the detailed solution:
1. Simplify the term inside the cube root:
- We have [tex]\( \sqrt[3]{x y^3} \)[/tex].
- Using the properties of exponents to distribute the cube root, we get:
[tex]\[
\sqrt[3]{x y^3} = (x y^3)^{\frac{1}{3}}
\][/tex]
[tex]\[
(x y^3)^{\frac{1}{3}} = x^{\frac{1}{3}} \cdot y
\][/tex]
2. Substitute back into the original expression:
- The original expression is:
[tex]\[
4 x^{\frac{2}{3}} y^4 \sqrt[3]{x y^3}
\][/tex]
- Substituting the simplified cube root term, we get:
[tex]\[
4 x^{\frac{2}{3}} y^4 (x^{\frac{1}{3}} y)
\][/tex]
3. Combine the exponents for [tex]\(x\)[/tex] and [tex]\(y\)[/tex]:
- First, combine the [tex]\(x\)[/tex] terms:
[tex]\[
x^{\frac{2}{3}} \cdot x^{\frac{1}{3}} = x^{\frac{2}{3} + \frac{1}{3}} = x^1 = x
\][/tex]
- Next, combine the [tex]\(y\)[/tex] terms:
[tex]\[
y^4 \cdot y = y^{4+1} = y^5
\][/tex]
4. Put it all together:
- After combining the terms, the expression is simplified to:
[tex]\[
4 x y^5
\][/tex]
5. Identify the coefficients and exponents:
- Now, the expression is in the form [tex]\(a x^b y^c\)[/tex], where:
- [tex]\(a = 4\)[/tex],
- [tex]\(b = 1\)[/tex],
- [tex]\(c = 5\)[/tex].
6. Calculate the product [tex]\(a \cdot b \cdot c\)[/tex]:
- The product of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] is:
[tex]\[
4 \cdot 1 \cdot 5 = 20
\][/tex]
Therefore, the product of [tex]\(a, b\)[/tex], and [tex]\(c\)[/tex] is [tex]\(\boxed{20}\)[/tex].
