To simplify the given expression [tex]\( x^3 \sqrt[4]{32 x^5 y^3} \)[/tex], we can follow these steps:
1. Understand the original expression:
[tex]\[
x^3 \sqrt[4]{32 x^5 y^3}
\][/tex]
2. Rewrite the expression inside the fourth root:
[tex]\[
\sqrt[4]{32 x^5 y^3}
\][/tex]
3. Factorize 32:
[tex]\[
32 = 2^5
\][/tex]
Therefore,
[tex]\[
\sqrt[4]{32 x^5 y^3} = \sqrt[4]{2^5 x^5 y^3}
\][/tex]
4. Combine the exponents inside the fourth root:
[tex]\[
\sqrt[4]{2^5 x^5 y^3} = \sqrt[4]{(2x)^5 y^3}
\][/tex]
5. Simplify using the property of roots:
[tex]\[
\sqrt[4]{(2x)^5 y^3} = \sqrt[4]{(2x)^5} \cdot \sqrt[4]{y^3}
\][/tex]
6. Separate the terms and simplify:
[tex]\[
\sqrt[4]{(2x)^5} = (2x)^{5/4}
\][/tex]
And,
[tex]\[
\sqrt[4]{y^3} = y^{3/4}
\][/tex]
7. Combine the simplified radical terms:
[tex]\[
x^3 \cdot (2x)^{5/4} \cdot y^{3/4}
\][/tex]
8. Rewrite in a single expression:
[tex]\[
x^3 \cdot 2^{5/4} \cdot x^{5/4} \cdot y^{3/4}
\][/tex]
9. Combine like terms by adding exponents:
[tex]\[
2^{5/4} \cdot x^{3+5/4} \cdot y^{3/4}
\][/tex]
10. Simplify the exponent:
[tex]\[
x^{3 + \frac{5}{4}} = x^{\frac{12}{4} + \frac{5}{4}} = x^{\frac{17}{4}}
\][/tex]
11. Combine everything in the final form:
[tex]\[
2^{5/4} \cdot x^{17/4} \cdot y^{3/4}
\][/tex]
To express as a multiplication with coefficients, convert [tex]\(2^{5/4}\)[/tex] known that [tex]\(2^{5/4} = 2 \cdot 2^{1/4}\)[/tex]:
[tex]\[
2 \cdot 2^{1/4} \cdot x^{17/4} \cdot y^{3/4}
\][/tex]
Therefore, the simplified form of the expression [tex]\(x^3 \sqrt[4]{32 x^5 y^3}\)[/tex] is:
[tex]\[
2 \cdot 2^{1/4} \cdot x^{17/4} \cdot y^{3/4}
\][/tex]
