Answer :
Let's organize the steps necessary to simplify the expression [tex]\( 3x^2y^2 \cdot (7xy^3)^{\frac{1}{4}} \)[/tex] using the properties of rational exponents. Here’s a step-by-step simplification process:
1. Start with the original expression:
[tex]\[ 3x^2y^2 \cdot \sqrt[4]{7xy^3} \][/tex]
Corresponding tile:
[tex]\[ 3 x^2 y^2 \sqrt[4]{7 x y^3} \][/tex]
2. Apply the exponent to each term inside the parentheses:
[tex]\[ 3x^2y^2 \cdot 7^{\frac{1}{4}} \cdot x^{\frac{1}{4}} \cdot y^{\frac{3}{4}} \][/tex]
Corresponding tile:
[tex]\[ 3 \cdot x^2 \cdot y^2 \cdot (7^{\frac{1}{4}} \cdot x^{\frac{1}{4}} \cdot y^{\frac{3}{4}}) \][/tex]
3. Combine the terms with the same bases:
[tex]\[ 3^1 \cdot 7^{\frac{1}{4}} \cdot x^{2 + \frac{1}{4}} \cdot y^{2 + \frac{3}{4}} \][/tex]
Corresponding tile:
[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 (simplify exponents):
[tex]\[ 3 \cdot 7^{\frac{1}{4}} \cdot x^{\frac{9}{4}} \cdot y^{\frac{11}{4}} \][/tex]
Corresponding tile:
[tex]\[ \left( 3^1 \cdot 7^{\frac{1}{4}} \cdot x^2 \cdot x^{\frac{1}{4}} \cdot y^2 \cdot y^{\frac{3}{4}} \right) \][/tex]
5. Express using properties of exponents and radicals:
[tex]\[ \left(3^4 \cdot 7 \cdot x^9 \cdot y^{11}\right)^{\frac{1}{4}} \][/tex]
Corresponding tile:
[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]
6. Combine the coefficients and variable terms under the radical sign:
[tex]\[ (81 \cdot 7 \cdot x^9 \cdot y^{11})^{\frac{1}{4}} \][/tex]
Corresponding tile:
[tex]\[ \left(81 \cdot 7\right)^{\frac{1}{4}} \cdot x^{\left(\frac{8}{4} + \frac{1}{4}\right)} \cdot y^{\left(\frac{8}{4} + \frac{3}{4}\right)} \][/tex]
7. Express the complete simplified term:
[tex]\[ \left(567 x^9 y^{11}\right)^{\frac{1}{4}} \][/tex]
Corresponding tile:
[tex]\[ \left(567 x^9 y^{11}\right)^{\frac{1}{4}} \][/tex]
8. Rewrite as a simplified radical expression:
[tex]\[ \sqrt[4]{567 x^9 y^{11}} \][/tex]
Corresponding tile:
[tex]\[ \sqrt[4]{567 x^9 y^{11}} \][/tex]
To summarize:
1. [tex]\( 3 x^2 y^2 \cdot \sqrt[4]{7 x y^3} \)[/tex]
2. [tex]\( 3 \cdot x^2 \cdot y^2 \cdot (7^{\frac{1}{4}} \cdot x^{\frac{1}{4}} \cdot y^{\frac{3}{4}}) \)[/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]\( \left( 3^1 \cdot 7^{\frac{1}{4}} \cdot x^2 \cdot x^{\frac{1}{4}} \cdot y^2 \cdot y^{\frac{3}{4}} \right) \)[/tex]
5. [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]
6. [tex]\( \left(81 \cdot 7\right)^{\frac{1}{4}} \cdot x^{\left(\frac{8}{4} + \frac{1}{4}\right)} \cdot y^{\left(\frac{8}{4} + \frac{3}{4}\right)} \)[/tex]
7. [tex]\( \left(567 x^9 y^{11}\right)^{\frac{1}{4}} \)[/tex]
8. [tex]\( \sqrt[4]{567 x^9 y^{11}} \)[/tex]
1. Start with the original expression:
[tex]\[ 3x^2y^2 \cdot \sqrt[4]{7xy^3} \][/tex]
Corresponding tile:
[tex]\[ 3 x^2 y^2 \sqrt[4]{7 x y^3} \][/tex]
2. Apply the exponent to each term inside the parentheses:
[tex]\[ 3x^2y^2 \cdot 7^{\frac{1}{4}} \cdot x^{\frac{1}{4}} \cdot y^{\frac{3}{4}} \][/tex]
Corresponding tile:
[tex]\[ 3 \cdot x^2 \cdot y^2 \cdot (7^{\frac{1}{4}} \cdot x^{\frac{1}{4}} \cdot y^{\frac{3}{4}}) \][/tex]
3. Combine the terms with the same bases:
[tex]\[ 3^1 \cdot 7^{\frac{1}{4}} \cdot x^{2 + \frac{1}{4}} \cdot y^{2 + \frac{3}{4}} \][/tex]
Corresponding tile:
[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 (simplify exponents):
[tex]\[ 3 \cdot 7^{\frac{1}{4}} \cdot x^{\frac{9}{4}} \cdot y^{\frac{11}{4}} \][/tex]
Corresponding tile:
[tex]\[ \left( 3^1 \cdot 7^{\frac{1}{4}} \cdot x^2 \cdot x^{\frac{1}{4}} \cdot y^2 \cdot y^{\frac{3}{4}} \right) \][/tex]
5. Express using properties of exponents and radicals:
[tex]\[ \left(3^4 \cdot 7 \cdot x^9 \cdot y^{11}\right)^{\frac{1}{4}} \][/tex]
Corresponding tile:
[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]
6. Combine the coefficients and variable terms under the radical sign:
[tex]\[ (81 \cdot 7 \cdot x^9 \cdot y^{11})^{\frac{1}{4}} \][/tex]
Corresponding tile:
[tex]\[ \left(81 \cdot 7\right)^{\frac{1}{4}} \cdot x^{\left(\frac{8}{4} + \frac{1}{4}\right)} \cdot y^{\left(\frac{8}{4} + \frac{3}{4}\right)} \][/tex]
7. Express the complete simplified term:
[tex]\[ \left(567 x^9 y^{11}\right)^{\frac{1}{4}} \][/tex]
Corresponding tile:
[tex]\[ \left(567 x^9 y^{11}\right)^{\frac{1}{4}} \][/tex]
8. Rewrite as a simplified radical expression:
[tex]\[ \sqrt[4]{567 x^9 y^{11}} \][/tex]
Corresponding tile:
[tex]\[ \sqrt[4]{567 x^9 y^{11}} \][/tex]
To summarize:
1. [tex]\( 3 x^2 y^2 \cdot \sqrt[4]{7 x y^3} \)[/tex]
2. [tex]\( 3 \cdot x^2 \cdot y^2 \cdot (7^{\frac{1}{4}} \cdot x^{\frac{1}{4}} \cdot y^{\frac{3}{4}}) \)[/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]\( \left( 3^1 \cdot 7^{\frac{1}{4}} \cdot x^2 \cdot x^{\frac{1}{4}} \cdot y^2 \cdot y^{\frac{3}{4}} \right) \)[/tex]
5. [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]
6. [tex]\( \left(81 \cdot 7\right)^{\frac{1}{4}} \cdot x^{\left(\frac{8}{4} + \frac{1}{4}\right)} \cdot y^{\left(\frac{8}{4} + \frac{3}{4}\right)} \)[/tex]
7. [tex]\( \left(567 x^9 y^{11}\right)^{\frac{1}{4}} \)[/tex]
8. [tex]\( \sqrt[4]{567 x^9 y^{11}} \)[/tex]
