To solve the given expression [tex]\(x^{-\frac{4}{3}} y \sqrt[3]{8 x^5 y^3}\)[/tex] and write it in the form [tex]\(a x^b y^c\)[/tex], and then determine the product of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex], we follow these steps:
1. Simplify the cube root expression:
[tex]\[
\text{Given expression: } x^{-\frac{4}{3}} y \sqrt[3]{8 x^5 y^3}
\][/tex]
First, let's simplify [tex]\(\sqrt[3]{8 x^5 y^3}\)[/tex].
2. Break down the cube root expression:
[tex]\[
\sqrt[3]{8 x^5 y^3} = \sqrt[3]{8} \cdot \sqrt[3]{x^5} \cdot \sqrt[3]{y^3}
\][/tex]
3. Evaluate each cube root:
[tex]\[
\sqrt[3]{8} = 2
\][/tex]
[tex]\[
\sqrt[3]{x^5} = x^{5/3}
\][/tex]
[tex]\[
\sqrt[3]{y^3} = y
\][/tex]
Therefore,
[tex]\[
\sqrt[3]{8 x^5 y^3} = 2 x^{5/3} y
\][/tex]
4. Combine the simplified cube root with the outer expression:
Substituting back into the original expression:
[tex]\[
x^{-\frac{4}{3}} y \cdot 2 x^{5/3} y = 2 x^{-\frac{4}{3}} x^{5/3} y \cdot y
\][/tex]
5. Combine the exponents of [tex]\(x\)[/tex]:
When multiplying powers of the same base, add the exponents:
[tex]\[
x^{-\frac{4}{3}} \cdot x^{\frac{5}{3}} = x^{-\frac{4}{3} + \frac{5}{3}} = x^{\frac{1}{3}}
\][/tex]
Therefore,
[tex]\[
2 x^{\frac{1}{3}} y \cdot y = 2 x^{\frac{1}{3}} y^2
\][/tex]
6. Identify the constants [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
The simplified expression is [tex]\(2 x^{1/3} y^2\)[/tex], where:
[tex]\[
a = 2, \quad b = \frac{1}{3}, \quad c = 2
\][/tex]
7. Calculate the product [tex]\(a \cdot b \cdot c\)[/tex]:
[tex]\[
a \cdot b \cdot c = 2 \cdot \frac{1}{3} \cdot 2 = \frac{4}{3}
\][/tex]
Therefore, the product of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] is [tex]\(\frac{4}{3}\)[/tex].
