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Drag each tile to the correct box. Order the simplification steps of the expression below using the properties of rational exponents.

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

[tex]\[
\begin{array}{l}
\square\left(567 x^9 y^{11}\right)^{\frac{1}{4}} \\
3 x^2 y^2 \sqrt[4]{7 x y^3} \\
3 x^2 y^2 \cdot\left(7 x y^3\right)^{\frac{1}{4}} \\
3 \cdot x^2 \cdot y^2 \cdot\left(7^{\frac{1}{4}} \cdot x^{\frac{1}{4}} \cdot y^{\frac{3}{4}}\right) \\
3^1 \cdot 7^{\frac{1}{4}} \cdot x^2 \cdot x^{\frac{1}{4}} \cdot y^2 \cdot y^{\frac{3}{4}} \\
\end{array}
\][/tex]

[tex]\[
\begin{array}{l}
\left(3^4\right)^{\frac{1}{4}} \cdot 7^{\frac{1}{4}} \cdot x^{\left(2+\frac{1}{4}\right)} \cdot y^{\left(2+\frac{3}{4}\right)} \\
(81)^{\frac{1}{4}} \cdot(7)^{\frac{1}{4}} \cdot x^{\left(\frac{8}{4}+\frac{1}{4}\right)} \cdot y^{\left(\frac{8}{4}+\frac{3}{4}\right)} \\
(81 \cdot 7)^{\frac{1}{4}} \cdot x^{\frac{2}{4}} \cdot y^{\frac{11}{4}} \\
\end{array}
\][/tex]


Sagot :

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.
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