To simplify the expression [tex]\(\sqrt[4]{567 x^9 y^{11}}\)[/tex] step by step using the properties of rational exponents, we can break it down as follows:
1. The original expression:
[tex]\[
\left(567 x^9 y^{11}\right)^{\frac{1}{4}}
\][/tex]
2. Break down 567 into its prime factors (81 and 7) and separate the exponents within the expression:
[tex]\[
\left((81 \cdot 7) x^8 x y^{8} y^{3}\right)^{\frac{1}{4}}
\][/tex]
3. Apply the exponent to each term inside the parentheses:
[tex]\[
(81)^{\frac{1}{4}} \cdot (7)^{\frac{1}{4}} \cdot (x^8)^{\frac{1}{4}} \cdot (x)^{\frac{1}{4}} \cdot (y^8)^{\frac{1}{4}} \cdot (y^3)^{\frac{1}{4}}
\][/tex]
4. Simplify the exponents:
[tex]\[
(81)^{\frac{1}{4}} \cdot (7)^{\frac{1}{4}} \cdot x^{\left(\frac{8}{4}\right)} \cdot x^{\frac{1}{4}} \cdot y^{\left(\frac{8}{4}\right)} \cdot y^{\left(\frac{3}{4}\right)}
\][/tex]
5. Combine like terms:
[tex]\[
\left(3^4\right)^{\frac{1}{4}} \cdot 7^{\frac{1}{4}} \cdot x^{\left(2+\frac{1}{4}\right)} \cdot y^{\left(2+\frac{3}{4}\right)}
\][/tex]
6. Evaluate the roots:
[tex]\[
3 \cdot (7)^{\frac{1}{4}} \cdot x^{\left(2+\frac{1}{4}\right)} \cdot y^{\left(2+\frac{3}{4}\right)}
\][/tex]
7. Present the simplified terms in a standard format:
[tex]\[
3 \cdot x^2 \cdot y^2 \cdot \left(7^{\frac{1}{4}} \cdot x^{\frac{1}{4}} \cdot y^{\frac{3}{4}}\right)
\][/tex]
8. Combine the terms into a more compact expression:
[tex]\[
3 \cdot x^2 \cdot y^2 \cdot \left(7 x y^3\right)^{\frac{1}{4}}
\][/tex]
Thus, the ordered simplification steps are:
1. [tex]\(\left(567 x^9 y^{11}\right)^{\frac{1}{4}}\)[/tex]
2. [tex]\(\left((81 \cdot 7) x^8 y^{8+3}\right)^{\frac{1}{4}}\)[/tex]
3. [tex]\((81)^{\frac{1}{4}} \cdot 7^{\frac{1}{4}} \cdot x^{\frac{8}{4}} \cdot y^{\frac{11}{4}}\)[/tex]
4. [tex]\(\left(3^4\right)^{\frac{1}{4}} \cdot 7^{\frac{1}{4}} \cdot x^{\left(2+\frac{1}{4}\right)} \cdot y^{\left(2+\frac{3}{4}\right)}\)[/tex]
5. [tex]\(3 \cdot (7)^{\frac{1}{4}} \cdot x^{\left(2+\frac{1}{4}\right)} \cdot y^{\left(2+\frac{3}{4}\right)}\)[/tex]
6. [tex]\(3 \cdot x^2 \cdot y^2 \cdot \left(7^{\frac{1}{4}} x^{\frac{1}{4}} y^{\frac{3}{4}}\right)\)[/tex]
7. [tex]\(3 \cdot x^2 \cdot y^2 \cdot \left(7 x y^3\right)^{\frac{1}{4}}\)[/tex]
8. [tex]\(3 x^2 y^2 \left(7 x y^3\right)^{\frac{1}{4}}\)[/tex]
