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Sagot :
Sure, I'll guide you step-by-step through the simplification process using the properties of rational exponents and the given expressions. We will order the steps correctly to reach the final simplified form.
1. Start with the initial expression:
[tex]\(\left(567 x^9 y^{11}\right)^{\frac{1}{4}}\)[/tex]
2. Factor the expression inside the parentheses to highlight perfect powers:
[tex]\(3 x^2 y^2 \cdot\left(7 x y^3\right)^{\frac{1}{4}}\)[/tex]
3. Separate the factors with their respective 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. Combine the exponents for [tex]\(x\)[/tex] and [tex]\(y\)[/tex]:
[tex]\(3^1 \cdot 7^{\frac{1}{4}} \cdot x^{\left(2+\frac{1}{4}\right)} \cdot y^{\left(2+\frac{3}{4}\right)}\)[/tex]
5. Simplify the combined exponents and factor out the outside parts:
[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]
Now, based on the given tiles the expressions and their order will be:
1. [tex]\(\left(567 x^9 y^{11}\right)^{\frac{1}{4}}\)[/tex]
2. [tex]\(3 x^2 y^2 \cdot\left(7 x y^3\right)^{\frac{1}{4}}\)[/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]\(\left(3^4\right)^{\frac{1}{4}} \cdot 7^{\frac{1}{4}} \cdot x^{\left(2+\frac{1}{4}\right)} \cdot y^{\left(2+\frac{3}{4}\right)}\)[/tex]
5. [tex]\(3 \cdot x^2 \cdot y^2 \cdot\left(7^{\frac{1}{4}} \cdot x^{\frac{1}{4}} \cdot y^{\frac{3}{4}}\right)\)[/tex]
So, aligning the steps correctly:
[tex]\[ \begin{aligned} 1. & \left(567 x^9 y^{11}\right)^{\frac{1}{4}}, \\ 2. & 3 x^2 y^2 \sqrt[4]{7 x y^3}, \\ 3. & 3^1 \cdot 7^{\frac{1}{4}} \cdot x^2 \cdot x^{\frac{1}{4}} \cdot y^2 \cdot y^{\frac{3}{4}}, \\ 4. & \left(3^4\right)^{\frac{1}{4}} \cdot 7^{\frac{1}{4}} \cdot x^{\left(2+\frac{1}{4}\right)} \cdot y^{\left(2+\frac{3}{4}\right)}, \\ 5. & 3 \cdot x^2 \cdot y^2 \cdot\left(7^{\frac{1}{4}} \cdot x^{\frac{1}{4}} \cdot y^{\frac{3}{4}}\right), \end{aligned} \][/tex]
Hence, the correct order will be:
1. [tex]\(\left(567 x^9 y^{11}\right)^{\frac{1}{4}}\)[/tex]
2. [tex]\(3 x^2 y^2 \cdot\left(7 x y^3\right)^{\frac{1}{4}}\)[/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]\(\left(3^4\right)^{\frac{1}{4}} \cdot 7^{\frac{1}{4}} \cdot x^{\left(2+\frac{1}{4}\right)} \cdot y^{\left(2+\frac{3}{4}\right)}\)[/tex]
5. [tex]\(3 \cdot x^2 \cdot y^2 \cdot\left(7^{\frac{1}{4}} \cdot x^{\frac{1}{4}} \cdot y^{\frac{3}{4}}\right)\)[/tex]
1. Start with the initial expression:
[tex]\(\left(567 x^9 y^{11}\right)^{\frac{1}{4}}\)[/tex]
2. Factor the expression inside the parentheses to highlight perfect powers:
[tex]\(3 x^2 y^2 \cdot\left(7 x y^3\right)^{\frac{1}{4}}\)[/tex]
3. Separate the factors with their respective 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. Combine the exponents for [tex]\(x\)[/tex] and [tex]\(y\)[/tex]:
[tex]\(3^1 \cdot 7^{\frac{1}{4}} \cdot x^{\left(2+\frac{1}{4}\right)} \cdot y^{\left(2+\frac{3}{4}\right)}\)[/tex]
5. Simplify the combined exponents and factor out the outside parts:
[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]
Now, based on the given tiles the expressions and their order will be:
1. [tex]\(\left(567 x^9 y^{11}\right)^{\frac{1}{4}}\)[/tex]
2. [tex]\(3 x^2 y^2 \cdot\left(7 x y^3\right)^{\frac{1}{4}}\)[/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]\(\left(3^4\right)^{\frac{1}{4}} \cdot 7^{\frac{1}{4}} \cdot x^{\left(2+\frac{1}{4}\right)} \cdot y^{\left(2+\frac{3}{4}\right)}\)[/tex]
5. [tex]\(3 \cdot x^2 \cdot y^2 \cdot\left(7^{\frac{1}{4}} \cdot x^{\frac{1}{4}} \cdot y^{\frac{3}{4}}\right)\)[/tex]
So, aligning the steps correctly:
[tex]\[ \begin{aligned} 1. & \left(567 x^9 y^{11}\right)^{\frac{1}{4}}, \\ 2. & 3 x^2 y^2 \sqrt[4]{7 x y^3}, \\ 3. & 3^1 \cdot 7^{\frac{1}{4}} \cdot x^2 \cdot x^{\frac{1}{4}} \cdot y^2 \cdot y^{\frac{3}{4}}, \\ 4. & \left(3^4\right)^{\frac{1}{4}} \cdot 7^{\frac{1}{4}} \cdot x^{\left(2+\frac{1}{4}\right)} \cdot y^{\left(2+\frac{3}{4}\right)}, \\ 5. & 3 \cdot x^2 \cdot y^2 \cdot\left(7^{\frac{1}{4}} \cdot x^{\frac{1}{4}} \cdot y^{\frac{3}{4}}\right), \end{aligned} \][/tex]
Hence, the correct order will be:
1. [tex]\(\left(567 x^9 y^{11}\right)^{\frac{1}{4}}\)[/tex]
2. [tex]\(3 x^2 y^2 \cdot\left(7 x y^3\right)^{\frac{1}{4}}\)[/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]\(\left(3^4\right)^{\frac{1}{4}} \cdot 7^{\frac{1}{4}} \cdot x^{\left(2+\frac{1}{4}\right)} \cdot y^{\left(2+\frac{3}{4}\right)}\)[/tex]
5. [tex]\(3 \cdot x^2 \cdot y^2 \cdot\left(7^{\frac{1}{4}} \cdot x^{\frac{1}{4}} \cdot y^{\frac{3}{4}}\right)\)[/tex]
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