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Order the simplification steps of the expression below using the properties of rational exponents.

[tex] \sqrt[4]{567 x^9 y^{11}} [/tex]

1. [tex] \left(567 x^9 y^{11}\right)^{\frac{1}{4}} [/tex]
2. [tex] (81 \cdot 7)^{\frac{1}{4}} \cdot x^{\frac{9}{4}} \cdot y^{\frac{11}{4}} [/tex]
3. [tex] (81)^{\frac{1}{4}} \cdot (7)^{\frac{1}{4}} \cdot x^{\left(\frac{8}{4}+\frac{1}{4}\right)} \cdot y^{\left(\frac{8}{4}+\frac{3}{4}\right)} [/tex]
4. [tex] (3^4)^{\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 x^2 \cdot y^2 \cdot \left(7^{\frac{1}{4}} \cdot x^{\frac{1}{4}} \cdot y^{\frac{3}{4}}\right) [/tex]
6. [tex] 3 x^2 y^2 \cdot \left(7 x y^3\right)^{\frac{1}{4}} [/tex]
7. [tex] 3 x^2 y^2 \sqrt[4]{7 x y^3} [/tex]

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Answer :

GinnyAnswer
Let's go through the steps in order to simplify the expression [tex]\(\sqrt[4]{567 x^9 y^{11}}\)[/tex]:

1. Start by expressing the entire expression under the 4th root as raised to the power of [tex]\(\frac{1}{4}\)[/tex]:

[tex]\[ \left(567 x^9 y^{11}\right)^{\frac{1}{4}} \][/tex]

2. Apply the [tex]\(\frac{1}{4}\)[/tex] power to each element inside the expression separately:

[tex]\[ (81 \cdot 7)^{\frac{1}{4}} \cdot x^{\frac{9}{4}} \cdot y^{\frac{11}{4}} \][/tex]

3. Split [tex]\(81\)[/tex] into [tex]\(3^4\)[/tex] and apply the [tex]\(\frac{1}{4}\)[/tex] power accordingly:

[tex]\[ (3^4)^{\frac{1}{4}} \cdot 7^{\frac{1}{4}} \cdot x^{\frac{9}{4}} \cdot y^{\frac{11}{4}} \][/tex]

4. Simplify [tex]\( (3^4)^{\frac{1}{4}} \)[/tex] to [tex]\(3\)[/tex]:

[tex]\[ 3 \cdot 7^{\frac{1}{4}} \cdot x^{\frac{9}{4}} \cdot y^{\frac{11}{4}} \][/tex]

5. Decompose [tex]\(x^{\frac{9}{4}}\)[/tex] and [tex]\(y^{\frac{11}{4}}\)[/tex] into integer and fractional parts:

[tex]\[ 3 \cdot (7^{\frac{1}{4}}) \cdot (x^{2} \cdot x^{\frac{1}{4}}) \cdot (y^{2} \cdot y^{\frac{3}{4}}) \][/tex]

6. Re-group the terms to make the expression clearer:

[tex]\[ 3 \cdot x^2 \cdot y^2 \cdot (7^{\frac{1}{4}} \cdot x^{\frac{1}{4}} \cdot y^{\frac{3}{4}}) \][/tex]

7. Combine the simplified terms:

[tex]\[ 3 \cdot x^2 \cdot y^2 \cdot \sqrt[4]{7 x y^3} \][/tex]

Putting the steps in sequential order:

1. [tex]\(\left(567 x^9 y^{11}\right)^{\frac{1}{4}}\)[/tex]
2. (81 \cdot 7)^{\frac{1}{4}} \cdot x^{\frac{9}{4}} \cdot y^{\frac{11}{4}}
3. (3^4)^{\frac{1}{4}} \cdot 7^{\frac{1}{4}} \cdot x^{\frac{9}{4}} \cdot y^{\frac{11}{4}}
4. (81)^{\frac{1}{4}} \cdot(7)^{\frac{1}{4}} \cdot x^{\left(\frac{8}{4}+\frac{1}{4}\right)} \cdot y^{\left(\frac{8}{4}+\frac{3}{4}\right)}
5. 3 \cdot x^2 \cdot y^2 \cdot (7^{\frac{1}{4}} \cdot x^{\frac{1}{4}} \cdot y^{\frac{3}{4}})
6. 3 \cdot x^2 \cdot y^2 \cdot \left( 7 x y^3 \right)^{\frac{1}{4}}

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