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]\((3^4)^{\frac{1}{4}} \cdot 7^{\frac{1}{4}} \cdot x^{(2+\frac{1}{4})} \cdot y^{(2+\frac{1}{4})}\)[/tex]
2. [tex]\((81)^{\frac{1}{4}} \cdot (7)^{\frac{1}{4}} \cdot x^{(\frac{3}{4}+\frac{1}{4})} \cdot y^{(\frac{8}{4}+\frac{3}{4})}\)[/tex]
3. [tex]\(3^1 \cdot 7^{\frac{1}{4}} \cdot x^2 \cdot x^{\frac{1}{4}} \cdot y^2 \cdot y^{\frac{3}{4}}\)[/tex]
4. [tex]\(3 \cdot x^2 \cdot y^2 \cdot (7 \cdot x^{\frac{1}{4}} \cdot y^{\frac{3}{4}})\)[/tex]
5. [tex]\(3 x^2 y^2 \sqrt[4]{7 x y^3}\)[/tex]



Answer :

Certainly! Here you go:

1. First, we rewrite the expression in terms of its components:
[tex]\[ 3 \cdot x^2 \cdot y^2 \cdot \sqrt[4]{7 \cdot x \cdot y^3} \][/tex]

2. Next, we rewrite the radicals using properties of rational exponents:
[tex]\[ 3 x^2 y^2 \sqrt[4]{7 x y^3} \][/tex]

3. Then, we express each component under the root as a product of terms with rational exponents:
[tex]\[ 3^1 \cdot 7^{\frac{1}{4}} \cdot x^2 \cdot x^{\frac{1}{4}} \cdot y^2 \cdot y^{\frac{3}{4}} \][/tex]

4. We combine the like terms in the expression:
[tex]\[ (3^4)^{\frac{1}{4}} \cdot 7^{\frac{1}{4}} \cdot x^{2 + \frac{1}{4}} \cdot y^{2 + \frac{3}{4}} \][/tex]

5. Finally, we simplify the exponents and terms:
[tex]\[ (81)^{\frac{1}{4}} \cdot(7)^{\frac{1}{4}} \cdot x^{\frac{9}{4}} \cdot y^{\frac{11}{4}} \][/tex]

These are the steps for simplifying the expression using the properties of rational exponents:

1. [tex]$3 \cdot x^2 \cdot y^2 \cdot\left(7\right.$[/tex] 故 [tex]$\left.\cdot x^{\frac{1}{4}} \cdot y^{\frac{3}{4}}\right)$[/tex]
2. [tex]$3 x^2 y^2 \sqrt[4]{7 x y^3}$[/tex]
3. [tex]$3^1 \cdot 7^{\frac{1}{4}} \cdot x^2 \cdot x^{\frac{1}{4}} \cdot y^2 \cdot y^{\frac{3}{4}}$[/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]$(81)^{\frac{1}{4}} \cdot(7)^{\frac{1}{4}} \cdot x^{\left(2 + \frac{1}{4}\right)} \cdot y^{\left(2 + \frac{\frac{3}{4}\right) + 3}{4}}$[/tex]