Let's order the steps of simplifying the given expression step-by-step using the properties of rational exponents.
Given expression:
[tex]\[
\sqrt[4]{567 x^9 y^{11}}
\][/tex]
### Step-by-Step Simplification:
1. Prime Factorization and Initial Breakdown:
[tex]\[
\sqrt[4]{567 x^9 y^{11}} = \sqrt[4]{3^4 \cdot 7 \cdot x^9 \cdot y^{11}}
\][/tex]
Here, we break 567 into its prime factors and separately consider the powers of [tex]\(x\)[/tex] and [tex]\(y\)[/tex].
2. Separate Fourth Roots:
[tex]\[
(3^4)^{\frac{1}{4}} \cdot (7)^{\frac{1}{4}} \cdot (x^9)^{\frac{1}{4}} \cdot (y^{11})^{\frac{1}{4}}
\][/tex]
3. Simplify Exponents:
[tex]\[
3 \cdot 7^{\frac{1}{4}} \cdot x^{\frac{9}{4}} \cdot y^{\frac{11}{4}}
\][/tex]
4. Rewriting Exponents:
[tex]\[
3 \cdot 7^{\frac{1}{4}} \cdot x^{2 + \frac{1}{4}} \cdot y^{2 + \frac{3}{4}}
\][/tex]
5. Combining Like Terms:
[tex]\[
3 \cdot x^2 \cdot y^2 \cdot (7^{\frac{1}{4}} \cdot x^{\frac{1}{4}} \cdot y^{\frac{3}{4}})
\][/tex]
6. Final Form:
[tex]\[
3 x^2 y^2 \sqrt[4]{7 x y^3}
\][/tex]
Now matching these steps with the given tiles, the proper order of tiles would be:
1. [tex]\[
\left(3^4\right)^{\frac{1}{4}} \cdot 7^{\frac{1}{4}} \cdot x^{\left(2+\frac{1}{4}\right)} \cdot y^{\left(2+\frac{1}{4}\right)}
\][/tex]
2. [tex]\[
3^1 \cdot 7^{\frac{1}{6}} \cdot x^2 \cdot x^{\frac{1}{4}} \cdot y^2 \cdot y^{\frac{3}{6}}
\][/tex]
3. [tex]\[
(81 \cdot 7)^{\frac{1}{4}} \cdot x^{\frac{9}{6}} \cdot y^{\frac{11}{4}}
\][/tex]
4. [tex]\[
(81)^{\frac{1}{4}} \cdot(7)^{\frac{1}{8}} \cdot x^{\left(\frac{8}{8}+\frac{1}{8}\right)} \cdot y^{\left(\frac{8}{8}+\frac{3}{6}\right)}
\][/tex]
5. [tex]\[
3 \cdot x^2 \cdot y^2 \cdot\left(7^{\frac{1}{4}} \cdot x^{\frac{1}{4}} \cdot y^{\frac{2}{4}}\right)
\][/tex]
6. [tex]\[
3 x^2 y^2 \cdot\left(7 x y^3\right)^{\frac{1}{2}}
\][/tex]
7. [tex]\[
3 x^2 y^2 \sqrt[4]{7 x y^3}
\][/tex]
Putting these tiles in this sequence will correctly simplify the given expression.
