Let’s start breaking down the simplification of the expression [tex]\(\sqrt[4]{567 x^9 y^{11}}\)[/tex] step-by-step using rational exponents:
1. Rewrite the expression using rational exponents:
[tex]\[
\left(567 x^9 y^{11}\right)^{\frac{1}{4}}
\][/tex]
This is the starting point where we express the 4th root as an exponent.
2. Break down the coefficients and apply the exponent:
[tex]\[
(3^4 \cdot 7)^{\frac{1}{4}} \cdot (x^9)^{\frac{1}{4}} \cdot (y^{11})^{\frac{1}{4}}
\][/tex]
We can represent 567 as [tex]\(3^4 \cdot 7\)[/tex] and apply the exponent [tex]\(\frac{1}{4}\)[/tex] to each term.
3. Simplify within the parentheses (applying the rational exponents):
[tex]\[
3^{4 \cdot \frac{1}{4}} \cdot 7^{\frac{1}{4}} \cdot x^{\frac{9}{4}} \cdot y^{\frac{11}{4}}
\][/tex]
Simplify [tex]\(3^4 \cdot \frac{1}{4}\)[/tex] to get [tex]\(3\)[/tex] and keep the other terms as they are.
4. Combine the simplified terms appropriately:
[tex]\[
3 \cdot 7^{\frac{1}{4}} \cdot x^{\frac{9}{4}} \cdot y^{\frac{11}{4}}
\][/tex]
We simplify [tex]\(3 \cdot 7^{\frac{1}{4}}\)[/tex] and split the exponents of [tex]\(x\)[/tex] and [tex]\(y\)[/tex].
5. Organize the terms to further simplify:
[tex]\[
3 \cdot x^2 \cdot y^2 \cdot \left(7 x y^3\right)^{\frac{1}{4}}
\][/tex]
Here we split [tex]\(x^{\frac{9}{4}}\)[/tex] into [tex]\(x^2 \cdot x^{\frac{1}{4}}\)[/tex] and [tex]\(y^{\frac{11}{4}}\)[/tex] into [tex]\(y^2 \cdot y^{\frac{3}{4}}\)[/tex].
6. Factor out the remaining terms inside the 4th root:
[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]
Simplify the expression inside the root by splitting the terms.
7. Combine and rewrite the final simplified form:
[tex]\[
3 \cdot 7^{\frac{1}{4}} \cdot x^{2 + \frac{1}{4}} \cdot y^{2 + \frac{3}{4}}
\][/tex]
Combine the terms [tex]\(x\)[/tex] and [tex]\(y\)[/tex] using the properties of exponents.
Therefore, the ordered simplification steps for the expression [tex]\(\sqrt[4]{567 x^9 y^{11}}\)[/tex] are:
1. [tex]\(\left(567 x^9 y^{11}\right)^{\frac{1}{4}}\)[/tex]
2. [tex]\(3 x^2 y^2 \sqrt[4]{7 x y^3}\)[/tex]
3. [tex]\(3 x^2 y^2 \cdot(7 x y^3)^{\frac{1}{4}}\)[/tex]
4. [tex]\(3 \cdot x^2 \cdot y^2 \cdot (7^{\frac{1}{4}} \cdot x^{\frac{1}{4}} \cdot y^{\frac{3}{4}})\)[/tex]
5. [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. [tex]\(3 \cdot 7^{\frac{1}{4}} \cdot x^{2 + \frac{1}{4}} \cdot y^{2 + \frac{3}{4}}\)[/tex]
These steps represent the thorough simplification process of the given expression using properties of rational exponents.
