Sure! Let's organize the steps to simplify the expression [tex]\( \sqrt[4]{567 x^9 y^{11}} \)[/tex] in the correct order:
1. Initial Expression with Rational Exponents:
[tex]\[
\sqrt[4]{567 x^9 y^{11}} \rightarrow (567 x^9 y^{11})^{\frac{1}{4}}
\][/tex]
2. Break Down [tex]\( 567 \)[/tex]:
[tex]\[
567 = 81 \times 7 \Rightarrow (81 \cdot 7 \cdot x^9 \cdot y^{11})^{\frac{1}{4}}
\][/tex]
3. Distribute the Exponent:
[tex]\[
(81)^{\frac{1}{4}} \cdot (7)^{\frac{1}{4}} \cdot x^{\left(\frac{9}{4}\right)} \cdot y^{\left(\frac{11}{4}\right)}
\][/tex]
4. Simplify [tex]\( (81)^{\frac{1}{4}} = 3 \)[/tex]:
[tex]\[
3 \cdot (7)^{\frac{1}{4}} \cdot x^{\left(2 + \frac{1}{4}\right)} \cdot y^{\left(2 + \frac{3}{4}\right)}
\][/tex]
5. Combine Exponents:
[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]
6. Further Simplify:
[tex]\[
3 \cdot x^2 \cdot y^2 \cdot \left(7 x y^3\right)^{\frac{1}{4}}
\][/tex]
7. Final Form with Radical:
[tex]\[
3 \cdot x^2 \cdot y^2 \cdot \sqrt[4]{7 x y^3}
\][/tex]
Now, match the steps to the given tiles:
- Tile 1: [tex]\(\left(3^4\right)^{\frac{1}{4}} \cdot 7^{\frac{1}{4}} \cdot x^{\left(2+\frac{1}{4}\right)} \cdot y^{\left(2+\frac{3}{4}\right)}\)[/tex] corresponds to Step 4.
- Tile 2: [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] corresponds to Step 5.
- Tile 3: [tex]\(3 \cdot x^2 \cdot y^2 \cdot\left(7 x y^3\right)^{\frac{1}{4}}\)[/tex] corresponds to Step 6.
- Tile 4: [tex]\(3 \cdot x^2 \cdot y^2 \sqrt[4]{7 x y^3}\)[/tex] corresponds to Step 7.
- Tile 5: [tex]\( (81)^{\frac{1}{4}} \cdot (7)^{\frac{1}{4}} \cdot x^{\left(\frac{4}{4}+\frac{1}{4}\right)} \cdot y^{\left(\frac{4}{4}+\frac{3}{4}\right)} \)[/tex] corresponds to Step 3.
- Tile 6: [tex]\( (81 \cdot 7)^{\frac{1}{4}} \cdot x^{\frac{9}{4}} \cdot y^{\frac{11}{4}} \)[/tex] corresponds to Step 2.
- Tile 7: [tex]\( \left(567 x^9 y^{11}\right)^{\frac{1}{4}} \)[/tex] corresponds to Step 1.
