Welcome to Westonci.ca, the Q&A platform where your questions are met with detailed answers from experienced experts. Discover precise answers to your questions from a wide range of experts on our user-friendly Q&A platform. Get precise and detailed answers to your questions from a knowledgeable community of experts on our Q&A platform.
Sagot :
Let's order the steps of simplifying the given expression step-by-step using the properties of rational exponents.
Given expression:
[tex]\[ \sqrt[4]{567 x^9 y^{11}} \][/tex]
### Step-by-Step Simplification:
1. Prime Factorization and Initial Breakdown:
[tex]\[ \sqrt[4]{567 x^9 y^{11}} = \sqrt[4]{3^4 \cdot 7 \cdot x^9 \cdot y^{11}} \][/tex]
Here, we break 567 into its prime factors and separately consider the powers of [tex]\(x\)[/tex] and [tex]\(y\)[/tex].
2. Separate Fourth Roots:
[tex]\[ (3^4)^{\frac{1}{4}} \cdot (7)^{\frac{1}{4}} \cdot (x^9)^{\frac{1}{4}} \cdot (y^{11})^{\frac{1}{4}} \][/tex]
3. Simplify Exponents:
[tex]\[ 3 \cdot 7^{\frac{1}{4}} \cdot x^{\frac{9}{4}} \cdot y^{\frac{11}{4}} \][/tex]
4. Rewriting Exponents:
[tex]\[ 3 \cdot 7^{\frac{1}{4}} \cdot x^{2 + \frac{1}{4}} \cdot y^{2 + \frac{3}{4}} \][/tex]
5. Combining Like Terms:
[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. Final Form:
[tex]\[ 3 x^2 y^2 \sqrt[4]{7 x y^3} \][/tex]
Now matching these steps with the given tiles, the proper order of tiles would be:
1. [tex]\[ \left(3^4\right)^{\frac{1}{4}} \cdot 7^{\frac{1}{4}} \cdot x^{\left(2+\frac{1}{4}\right)} \cdot y^{\left(2+\frac{1}{4}\right)} \][/tex]
2. [tex]\[ 3^1 \cdot 7^{\frac{1}{6}} \cdot x^2 \cdot x^{\frac{1}{4}} \cdot y^2 \cdot y^{\frac{3}{6}} \][/tex]
3. [tex]\[ (81 \cdot 7)^{\frac{1}{4}} \cdot x^{\frac{9}{6}} \cdot y^{\frac{11}{4}} \][/tex]
4. [tex]\[ (81)^{\frac{1}{4}} \cdot(7)^{\frac{1}{8}} \cdot x^{\left(\frac{8}{8}+\frac{1}{8}\right)} \cdot y^{\left(\frac{8}{8}+\frac{3}{6}\right)} \][/tex]
5. [tex]\[ 3 \cdot x^2 \cdot y^2 \cdot\left(7^{\frac{1}{4}} \cdot x^{\frac{1}{4}} \cdot y^{\frac{2}{4}}\right) \][/tex]
6. [tex]\[ 3 x^2 y^2 \cdot\left(7 x y^3\right)^{\frac{1}{2}} \][/tex]
7. [tex]\[ 3 x^2 y^2 \sqrt[4]{7 x y^3} \][/tex]
Putting these tiles in this sequence will correctly simplify the given expression.
Given expression:
[tex]\[ \sqrt[4]{567 x^9 y^{11}} \][/tex]
### Step-by-Step Simplification:
1. Prime Factorization and Initial Breakdown:
[tex]\[ \sqrt[4]{567 x^9 y^{11}} = \sqrt[4]{3^4 \cdot 7 \cdot x^9 \cdot y^{11}} \][/tex]
Here, we break 567 into its prime factors and separately consider the powers of [tex]\(x\)[/tex] and [tex]\(y\)[/tex].
2. Separate Fourth Roots:
[tex]\[ (3^4)^{\frac{1}{4}} \cdot (7)^{\frac{1}{4}} \cdot (x^9)^{\frac{1}{4}} \cdot (y^{11})^{\frac{1}{4}} \][/tex]
3. Simplify Exponents:
[tex]\[ 3 \cdot 7^{\frac{1}{4}} \cdot x^{\frac{9}{4}} \cdot y^{\frac{11}{4}} \][/tex]
4. Rewriting Exponents:
[tex]\[ 3 \cdot 7^{\frac{1}{4}} \cdot x^{2 + \frac{1}{4}} \cdot y^{2 + \frac{3}{4}} \][/tex]
5. Combining Like Terms:
[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. Final Form:
[tex]\[ 3 x^2 y^2 \sqrt[4]{7 x y^3} \][/tex]
Now matching these steps with the given tiles, the proper order of tiles would be:
1. [tex]\[ \left(3^4\right)^{\frac{1}{4}} \cdot 7^{\frac{1}{4}} \cdot x^{\left(2+\frac{1}{4}\right)} \cdot y^{\left(2+\frac{1}{4}\right)} \][/tex]
2. [tex]\[ 3^1 \cdot 7^{\frac{1}{6}} \cdot x^2 \cdot x^{\frac{1}{4}} \cdot y^2 \cdot y^{\frac{3}{6}} \][/tex]
3. [tex]\[ (81 \cdot 7)^{\frac{1}{4}} \cdot x^{\frac{9}{6}} \cdot y^{\frac{11}{4}} \][/tex]
4. [tex]\[ (81)^{\frac{1}{4}} \cdot(7)^{\frac{1}{8}} \cdot x^{\left(\frac{8}{8}+\frac{1}{8}\right)} \cdot y^{\left(\frac{8}{8}+\frac{3}{6}\right)} \][/tex]
5. [tex]\[ 3 \cdot x^2 \cdot y^2 \cdot\left(7^{\frac{1}{4}} \cdot x^{\frac{1}{4}} \cdot y^{\frac{2}{4}}\right) \][/tex]
6. [tex]\[ 3 x^2 y^2 \cdot\left(7 x y^3\right)^{\frac{1}{2}} \][/tex]
7. [tex]\[ 3 x^2 y^2 \sqrt[4]{7 x y^3} \][/tex]
Putting these tiles in this sequence will correctly simplify the given expression.
Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. Westonci.ca is committed to providing accurate answers. Come back soon for more trustworthy information.
