To systematically simplify the expression [tex]\(\sqrt[3]{875 x^5 y^9}\)[/tex] using the properties of rational exponents, follow these detailed steps:
1. Start with the given expression:
[tex]\[
\left(875 x^5 y^9\right)^{\frac{1}{3}}
\][/tex]
2. Rewrite [tex]\(875\)[/tex] as the product of its factors:
[tex]\[
(125 \cdot 7)^{\frac{1}{3}} \cdot x^{\frac{5}{3}} \cdot y^{\frac{9}{3}}
\][/tex]
3. Express [tex]\(125\)[/tex] as [tex]\(5^3\)[/tex]:
[tex]\[
(125)^{\frac{1}{3}} \cdot (7)^{\frac{1}{3}} \cdot x^{\frac{5}{3}} \cdot y^{\frac{9}{3}}
\][/tex]
4. Simplify [tex]\((125)^{\frac{1}{3}}\)[/tex]:
[tex]\[
\left(5^3\right)^{\frac{1}{3}} \cdot 7^{\frac{1}{3}} \cdot x^{\left(\frac{3}{3} + \frac{2}{3}\right)} \cdot y^3
\][/tex]
5. Simplify [tex]\(\left(5^3\right)^{\frac{1}{3}}\)[/tex] to get [tex]\(5\)[/tex]:
[tex]\[
5 \cdot 7^{\frac{1}{3}} \cdot x^{1 + \frac{2}{3}} \cdot y^3
\][/tex]
6. Combine the exponents of [tex]\(x\)[/tex]:
[tex]\[
5 \cdot 7^{\frac{1}{3}} \cdot x x^{\frac{2}{3}} \cdot y^3
\][/tex]
7. Combine the similar terms:
[tex]\[
5 x y^3 \cdot \left(7 x^2\right)^{\frac{1}{3}}
\][/tex]
8. Rewrite the final simplified expression:
[tex]\[
5 x y^3 \sqrt[3]{7 x^2}
\][/tex]
So the ordered steps in the simplification are:
1. [tex]\(\left(875 x^5 y^9\right)^{\frac{1}{3}}\)[/tex]
2. (125 \cdot 7)^{\frac{1}{3}} \cdot x^{\frac{5}{3}} \cdot y^{\frac{9}{3}}
3. (125)^{\frac{1}{3}} \cdot (7)^{\frac{1}{3}} \cdot x^{\left(\frac{3}{3} + \frac{2}{3}\right)} \cdot y^3
4. \left(5^3\right)^{\frac{1}{3}} \cdot 7^{\frac{1}{3}} \cdot x^{\left(1 + \frac{2}{3}\right)} \cdot y^3
5. 5 \cdot 7^{\frac{1}{3}} \cdot x \cdot x^{\frac{2}{3}} \cdot y^3
6. 5 x y^3 (7 x^2)^{\frac{1}{3}}
7. 5 x y^3 \sqrt[3]{7 x^2}
This sequence orders the simplifications comprehensively, ensuring a step-by-step transformation to the final simplified form.
